Integrating computers and problem posing in mathematics teacher education 9789813273917, 9813273917

Table of contents :
Contents
Preface
Chapter 1 On the Genesis of Problem Posing in Mathematics
1.1 Problems from the first printed arithmetic
1.1.1 Solving a 15th century problem using the modern-day pedagogy
1.1.2 Posing 15th century-like problems through conceptualization
1.1.3 Using technology for posing 15th century-like problems
1.2 From a classic problem to using the modern spreadsheet
1.2.1 The birth of the probability theory through problem posing
1.2.2 Using a spreadsheet as a problem-posing tool
1.2.3 Duality of the spreadsheet’s use
1.3 The Problem of the Grand Duke of Tuscany
1.4 Conjecturing as posing problems to find proof
1.5 Problem posing in a classic context as a springboard into experimental mathematics
1.5.1 Triangular numbers with identical digits
1.5.2 Triangular number sieves
1.6 Problem posing as setting up a research program
1.7 Summary
Chapter 2 From a Theory of Problem Posing to Classroom Practice of the Digital Era
2.1 Problem posing as educational philosophy
2.2 Problem posing in the modern educational context
2.3 Learning to ask questions about posed/solved problems
2.4 Technology as a cultural support of problem posing
2.5 Numerical coherence in problem posing
2.5.1 Using a spreadsheet to pose a numerically coherent problem
2.6 Contextual coherence in problem posing
2.7 Pedagogical coherence in problem posing
2.8 Didactical coherence in problem posing
2.9 Summary
Chapter 3 Posing Technology-Immune/Technology-Enabled (TITE) Problems
3.1 From teaching machine movement to symbolic computations
3.2 Technological advances call for the revision of mathematics curriculum
3.3 Definition of a TITE problem and a simple example
3.4 Revisiting classic problems in the digital era under the umbrella of the TITE concept
3.5 Conceptual bond and arithmetical word problems
3.5.1 Looking at the past to develop new teaching ideas
3.5.2 Posing similar problems
3.6 Revisiting mathematical problems to make them didactically coherent
3.6.1 From numerical to contextual coherence
3.6.2 Towards pedagogical coherence
3.7 From modeling data to a general formula using technology
3.8 Formulating and solving a didactically coherent problem
3.9 Maple-based mathematical induction proof
3.10 Summary
Chapter 4 Linking Algorithmic Thinking and Conceptual Knowledge through Problem Posing
4.1 On the hierarchy of two types of knowledge
4.2 A simple question leads to revealing hidden creativity
4.3 Two levels of conceptual understanding
4.4 Solving a problem seeking information
4.5 Problem posing leads to conceptual knowledge and collateral learning
4.6 Using conceptual bond in posing problems with technology
4.7 Summary
Chapter 5 Using Graphing Software for Posing Problems in Advanced High School Algebra
5.1 Introduction
5.2 Location of roots of quadratics about an interval
5.3 Digital fabrication
5.4 Connecting the coordinate plane with the plane of coefficients
5.4.1 The case RREE
5.4.2 The case RERE
5.4.3 The case REER
5.4.4 The case ERER
5.4.5 The case EERR
5.4.6 The case ERRE
5.5 Using Vieta’s Theorem
5.6 Posing TITE problems in the plane of parameters
5.7 Geometric probabilities and the partitioning diagram
5.8 Making mathematical connections
5.9 Revealing hidden concepts through collateral learning
5.10 Summary
Chapter 6 Einstellung Effect and Problem Posing
6.1 Examples of Einstellung effect
6.2 Water jar experiments and Einstellung effect
6.3 Posing and solving problems as a remediation of Einstellung effect
6.4 Posing problems for water jar experiments using a spreadsheet
6.5 Einstellung effect in finding areas on a geoboard
6.6 Einstellung effect in solving algebraic equations and inequalities
6.7 Einstellung effect in solving trigonometric inequalities
6.8 Using technology to pose problems that might lead to Einstellung effect
6.9 Einstellung effect in solving logarithmic inequalities
6.9.1 Simultaneous extension and contraction of solution set
6.9.2 Extension of solution set
6.10 Solving logarithmic inequality (6.21) in the general case
6.10.1 The case n = 2k
6.10.2 The case n = 2k + 1
6.11 Summary
Chapter 7 Explorations with Integer Sequences as TITE Problem Posing
7.1 Introduction
7.2 Exploring patterns formed by the last digits of the sums of powers of integers
7.3 Discovering patters in the last digits of the polygonal numbers
7.3.1 The triangular number sieves
7.3.2 Triangular number sieves and the last digits of their terms
7.3.3 Rises and falls in permutations
7.3.4 Connecting triangular and square numbers within the multiplication table
7.3.5 The square number sieves
7.3.6 The pentagonal number sieves
7.3.7 The general case of the m-gonal number sieves
7.4 Patterns in the behavior of the greatest common divisors of two polygonal numbers
7.5 Exploring sequences formed by the sums of powers of integers
7.6 Exploring sieves developed from the sums of powers of integers
7.7 Summary
Appendix
8.1 Spreadsheets included in Chapter 1
8.2 Spreadsheets included in Chapter 2
8.3 Spreadsheets included in Chapter 3
8.4 Spreadsheets included in Chapter 4
8.5 Spreadsheets included in Chapter 6
8.6 Spreadsheets included in Chapter 7
Bibliography
Index

Citation preview

Other books by Sergei Abramovich

Exploring Mathematics with Integrated Spreadsheets in Teacher Education ISBN: 978-981-4678-22-3 Diversifying Mathematics Teaching — Advanced Educational Content and Methods for Prospective Elementary Teachers ISBN: 978-981-3206-87-8

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Cover design by Leonard Abramovich.

INTEGRATING  COMPUTERS  A ND  PROBLEM  POSING  IN  MATHEMATICS TEACHER  EDUCATION Copyright © 2019 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-3273-91-7

For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11089#t=suppl

Printed in Singapore

Preface This book is written to share ideas about integrating mathematical problem posing with the use of computing technology in the context of K–12 mathematics teacher preparation. Problem posing, in general, has been on the mathematics education agenda for a long time. It goes back to the 15th century Italy when the first printed book on arithmetic included real-life problems formulated with the goal to explain how the rules of arithmetic work in trade. Since then, over the centuries, mathematical problem posing appeared in different didactic forms as a way of enriching one’s learning experience through investigating mathematical ideas, exploring conjectures, and solving worthwhile problems. In the digital era, mathematics teacher education may include learning the skill of formulating different mathematical problems that the appropriate use of technology affords. This practical skill when augmented with conceptual understanding of problem posing can be referred to as an art of mathematics curriculum design. Conceptually, as shown in the book, the art is supported by two major theoretical positions that stem from technology integration: didactical coherence of a posed problem (Chapter 2) and technology-immune/technology-enabled (TITE) problem posing (Chapter 3). In order to connect theory and practice of mathematical problem posing in the digital era, the book includes examples of problems posed by teacher candidates enrolled in different technology-rich mathematics education courses taught by the author over the years. These examples are analyzed through the lenses of the proposed theory. In addition, the book shows how technology can be used to reformulate rather advanced problems from the traditional (pre-digital era) problem-solving curriculum. The goal of reformulation of such problems is at least twofold: to make them compatible with the modern-day emphasis on democratizing mathematics education, and to find the right balance between positive and negative affordances of technology. In particular, an argument can be made that through achieving such a balance one uses technology appropriately. v

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In the context of the book, the word technology refers to commonly available software tools including an electronic spreadsheet and various computer algebra and dynamic geometry systems such as Wolfram Alpha developed by Wolfram Research [www.wolframalpha.com], Maple [Char et al., 1991], the Graphing Calculator produced by Pacific Tech [Avitzur, 2011], and The Geometer’s Sketchpad [Jackiw, 1991]. This kind of combination of software tools for problem posing has been used by the author through teaching and research. The word appropriate refers to the fact that the tools of technology cannot be directly utilized for mathematical problem posing. Rather, such utilization requires one’s appreciation of their hidden educational capabilities as well as instructional pitfalls. In other words, the appropriate use of technology can be conceptualized as maximizing and minimizing, respectively, positive and negative affordances of the tools available in the digital era. The ideas behind this book stem from the author’s work with American and Canadian teacher candidates within different technologyrich mathematics education courses. The context of problem-posing activities shared by the author includes K-12 pre-service teacher education courses and activities with elementary school students administered by teacher candidates as part of their fieldwork (internship). The book shows that teacher candidates’ success with technology-enabled problem posing in the context of fostering creative thinking skills in the students of mathematics requires experience with mathematical modeling and problem solving on the part of the candidates. Furthermore, one can argue that technology-enabled problem posing provides teacher candidates with research-like experience in mathematics and leaves them with the sense of ownership of grade-appropriate extensions of the traditional mathematical content. The book consists of seven chapters and Appendix. The first chapter titled On the Genesis of Problem Posing in Mathematics describes the development of mathematical problem posing in the time span from the utilization of arithmetic in trade in the 15th century to the 20th century when problem posing was conceptualized by progressive educators as educational philosophy. In particular, problems that were posed in the 16th–17th centuries to decide the fair division of stakes in an unfinished game and to explain mathematics behind gambling observations are included.

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Out of those problems the theory of probability has grown. Problem posing is also discussed under the umbrella of setting up a research program for mathematics at the dawn of the 20th century. The use of the modern spreadsheet is shown as a milieu for posing problems similar to the above-mentioned problems from the classic mathematical heritage of humankind. The focus of the second chapter titled From a Theory of Problem Posing to Classroom Practice of the Digital Era is on the merit of problem posing as educational philosophy and the importance of didactic coherence in problem posing. In the context of problem posing, the chapter integrates philosophical, psychological, and educational ideas by John Dewey (reflective inquiry and collateral learning), Paulo Freire (learning by linking past and future developments), William James (motivating students’ interest in learning), Tsunesaburo Makiguchi (nurturing feelings of pleasure and happiness through education), Maria Montessori (liberty of a child as the spirit of education), and Lev Vygotsky (mediating learning by physical and psychological tools) to mathematics-inspired ideas by Archimedes (genesis of experimental mathematics), David Hilbert (providing clearness and ease of comprehension of ideas), and Benoit Mandelbrot (scaffolding the necessity of learning with the joy of knowing). The issue of didactic coherence is considered in the context of using computing technology when its negative affordances have to be minimized. To this end, three interrelated concepts – numerical coherence, contextual coherence, and pedagogical coherence – have been introduced as the major elements of didactical coherence of a problem. The three coherences are illustrated through the lenses of mathematical problems posed by prospective elementary teachers using a spreadsheet-based problem-posing environment designed by the author. The third chapter titled Posing Technology-Immune/TechnologyEnabled (TITE) Problems begins with a brief history of automated instruction and its evolution towards symbolic computations. It shows how the advent of computing technology into the classroom gave birth to new ideas about problem posing. Through the above-mentioned conceptualization of the appropriate use of technology, the notion of technology-immune/technology-enabled (TITE) problem has come to light. Such a problem cannot be automatically solved by software; yet

viii Integrating Computers and Problem Posing in Mathematics Teacher Education

educational computing is an essential part of a problem-solving process. The chapter revisits different problems of the pre-digital era borrowed from authors as different as George Pólya – one of the major contributors to mathematics education in the 20th century, and Anton Pavlovich Tchekoff – one of the best short story writers of all time. Drawing on these classic sources, the chapter suggests several modifications to the problems towards the development of TITE mathematics curriculum. Also, using the concept of didactic coherence of a problem, the chapter discusses ways of posing such problems with support of technology. In Chapter 4 titled Linking Algorithmic Thinking and Conceptual Knowledge through Problem Posing it is suggested that problem posing can be used as a link between procedural skills and conceptual knowledge. With this in mind, an argument is made that solving and posing a mathematical problem requires two levels of conceptual understanding – basic and advanced. The interplay between the two levels is analyzed and the use of technology in bridging the procedural and the conceptual is discussed. The goal of this discussion is the appreciation of the role that the cycle “solve-reflect-pose” plays in bridging procedural skills as part of basic conceptual understanding and conceptual knowledge as the core of advanced conceptual understanding. Through this cycle, what is considered advanced conceptual understanding at one cognitive process level turns into the basic conceptual understanding at a higher level. Likewise, the chapter distinguishes between the first-order questions (seeking information) and the second-order questions (requesting explanation). Using simple examples, it is demonstrated that although algebra (as “generalized arithmetic” [Philipp and Schappelle, 1999]) belongs to a higher cognitive process level than arithmetic, a traditional algebraic solution requires only basic conceptual understanding (in order to answer the first-order questions) and a purely arithmetic solution requires advanced conceptual understanding (in order to answer the second-order questions). The importance of seeing mathematical problem posing as a recurrent reflection on a method used to solve a problem is emphasized throughout the chapter. Chapter 5 titled Using Graphing Software for Posing Problems in Advanced High School Algebra shows how secondary teacher candidates can turn a mundane context of finding roots of quadratic equations into

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open-ended mathematical explorations rudimental to real mathematical problems arising in the disciplines of science and engineering. It presents technology as an enabler of posing exploratory problems dealing with the location of real roots of quadratics with two real parameters about an interval on the number line. TITE explorations stemming from self-posed problems provide secondary teacher candidates with research-like experience in formal mathematics and its technology-enabled pedagogy. As a rich milieu of problem posing, diagrams in the plane of parameters constructed through the joint use of visual and analytic representations of the concepts involved, have been used. These diagrams, responsible for a particular type of roots’ location, enable alternative queries dealing with geometric probabilities. The chapter emphasizes mathematical connections and historical perspectives. Under the umbrella of collateral learning, the notion of hidden inequalities is discussed. Chapter 6 titled Einstellung Effect and Problem Posing reveals the cases of Einstellung effect – a psychological phenomenon when successful experience with a certain type of problems leads to automatic problemsolving behavior which is either erroneous in a new conceptual domain, or simply lacks insight (alternatively, productive thinking). The chapter begins with examples of incorrect use of proportional reasoning and shows how educational computing can help one turn an erroneous thought into a thinking device. The use of a spreadsheet as a problem-posing environment is demonstrated in the context of classic water jar experiments that were used by Gestalt psychologists towards the study of productive thinking and ways of the remediation of Einstellung effect. Finding areas on a geoboard is discussed under the umbrella of reification – one of the main principles of Gestalt psychology dealing with the constructive aspect of perception which allows one to use insight in order to recognize in an image a seemingly hidden information. It is suggested that by bridging procedural skills and conceptual knowledge, negative affordances of Einstellung effect in mathematical problem solving can be reduced. Such a bridge can be built through the seamless integration of problem solving and problem posing. Polynomial, trigonometric, and logarithmic equations and inequalities in one variable are considered in order to reveal the sources of Einstellung effect caused by their erroneous simplification. Computer graphing software is shown as an appropriate

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environment for posing one-variable equations and inequalities when the cancellation of common variable factor leads to specific outcomes such as extension and/or contraction of the solution set of the original equation/inequality. In Chapter 7 titled Explorations with Integer Sequences as TITE Problem Posing mathematical activities developed by the author are included as examples of how technology can be utilized to enable its appropriate use (that is, creatively balancing its positive and negative affordances) in the modern-day classroom of prospective secondary mathematics teachers. TITE problems suggested in this chapter are motivated by a task from an (advanced by design) problem-solving book requiring one to find the last digit of a complicated numeric expression comprised of the sums of fifth powers of integers. Nowadays the task can be easily completed by plugging this expression into the input box of a computer algebra system, e.g., Wolfram Alpha, which immediately generates its value allowing one to see all the digits of the number, including the last digit. As shown in this chapter, the ease of computing of complicated numeric expressions, like the sums of powers of integers, makes it possible, by changing various characteristics of the sums, to recognize interesting patterns in the behavior of the last digits of the sums in question. This motivates extending explorations to other types of integer sequences such as polygonal numbers. In the context of this extension the notion of the polygonal number sieve of order k is introduced and explored for triangular, square, and pentagonal numbers from a combinatorial perspective using the concept of rises and falls in permutations. The chapter also extends explorations, briefly mentioned in Chapter 3, to include patterns in the behavior of the greatest common divisors of several polygonal number sieves. The chapter concludes with exploring number sieves developed from the sums of powers of integers. A technologyimmune part of these explorations consists of developing closed formulas for sequences not included in the Online Encyclopedia of Integer Sequences (OEIS), a rich source of mathematical knowledge in the digital era. Appendix is written to provide details of programing of spreadsheet environments developed by the author to support various problem-posing activities described in the book. Syntactic versatility of spreadsheets

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allows for alternative programming of computationally identical environments by modifying spreadsheet formulas presented in Appendix. Just as any reference to information available in a non-traditional print is accurate as of the day of its on-line retrieval, everything in Appendix is related to Microsoft Excel for Mac 2017, version 16.10. In conclusion, with much respect I wish to express my sincere gratitude to Ms Rok Ting Tan for encouraging me to submit a book proposal to World Scientific that, with deeply appreciated support and recommendations of anonymous reviewers, resulted in the present publication. Thanks are due to Ms Yolande Koh for her editorial guidance during my final work on the book. Last but not least, I acknowledge being under obligation to my son Leonard Abramovich for his professionalism in the design of the cover of the book.

Sergei Abramovich Potsdam, NY

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Contents Preface

v

Chapter 1 On the Genesis of Problem Posing in Mathematics ............................. 1.1 Problems from the first printed arithmetic ............................ 1.1.1 Solving a 15th century problem using the modern-day pedagogy .................................................................... 1.1.2 Posing 15th century-like problems through conceptualization ........................................................ 1.1.3 Using technology for posing 15th century-like problems ..................................................................... 1.2 From a classic problem to using the modern spreadsheet ..... 1.2.1 The birth of the probability theory through problem posing ......................................................................... 1.2.2 Using a spreadsheet as a problem-posing tool ........... 1.2.3 Duality of the spreadsheet’s use ................................. 1.3 The Problem of the Grand Duke of Tuscany ........................ 1.4 Conjecturing as posing problems to find proof ..................... 1.5 Problem posing in a classic context as a springboard into experimental mathematics..................................................... 1.5.1 Triangular numbers with identical digits .................... 1.5.2 Triangular number sieves ........................................... 1.6 Problem posing as setting up a research program ................. 1.7 Summary ...............................................................................

19 20 20 22 25

Chapter 2 From a Theory of Problem Posing to Classroom Practice of the Digital Era ............................................................................................ 2.1 Problem posing as educational philosophy ........................... 2.2 Problem posing in the modern educational context .............. 2.3 Learning to ask questions about posed/solved problems ......

27 27 29 32

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1 1 3 5 6 8 8 11 13 14 17

xiv Integrating Computers and Problem Posing in Mathematics Teacher Education

2.4 Technology as a cultural support of problem posing ............ 2.5 Numerical coherence in problem posing .............................. 2.5.1 Using a spreadsheet to pose a numerically coherent problem....................................................................... 2.6 Contextual coherence in problem posing .............................. 2.7 Pedagogical coherence in problem posing ............................ 2.8 Didactical coherence in problem posing ............................... 2.9 Summary ............................................................................... Chapter 3 Posing Technology-Immune/Technology-Enabled (TITE) Problems . 3.1 From teaching machine movement to symbolic computations ......................................................................... 3.2 Technological advances call for the revision of mathematics curriculum ............................................................................. 3.3 Definition of a TITE problem and a simple example ........... 3.4 Revisiting classic problems in the digital era under the umbrella of the TITE concept ............................................... 3.5 Conceptual bond and arithmetical word problems................ 3.5.1 Looking at the past to develop new teaching ideas .... 3.5.2 Posing similar problems ............................................. 3.6 Revisiting mathematical problems to make them didactically coherent ............................................................. 3.6.1 From numerical to contextual coherence ................... 3.6.2 Towards pedagogical coherence................................. 3.7 From modeling data to a general formula using technology 3.8 Formulating and solving a didactically coherent problem ... 3.9 Maple-based mathematical induction proof ......................... 3.10 Summary ............................................................................... Chapter 4 Linking Algorithmic Thinking and Conceptual Knowledge through Problem Posing ....................................................................... 4.1 On the hierarchy of two types of knowledge ........................ 4.2 A simple question leads to revealing hidden creativity ........ 4.3 Two levels of conceptual understanding ...............................

35 36 38 41 44 48 49

51 51 54 58 60 67 67 68 71 71 73 74 75 78 82

85 85 89 92

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4.4 Solving a problem seeking information ................................ 4.5 Problem posing leads to conceptual knowledge and collateral learning.................................................................. 4.6 Using conceptual bond in posing problems with technology 4.7 Summary ...............................................................................

93

Chapter 5 Using Graphing Software for Posing Problems in Advanced High School Algebra............................................................................ 5.1 Introduction ........................................................................... 5.2 Location of roots of quadratics about an interval.................. 5.3 Digital fabrication ................................................................. 5.4 Connecting the coordinate plane with the plane of coefficients........................................................................ 5.4.1 The case RREE ........................................................... 5.4.2 The case RERE ........................................................... 5.4.3 The case REER ........................................................... 5.4.4 The case ERER ........................................................... 5.4.5 The case EERR ........................................................... 5.4.6 The case ERRE ........................................................... 5.5 Using Vieta’s Theorem ......................................................... 5.6 Posing TITE problems in the plane of parameters ................ 5.7 Geometric probabilities and the partitioning diagram .......... 5.8 Making mathematical connections........................................ 5.9 Revealing hidden concepts through collateral learning ........ 5.10 Summary ............................................................................... Chapter 6 Einstellung Effect and Problem Posing ............................................... 6.1 Examples of Einstellung effect ............................................. 6.2 Water jar experiments and Einstellung effect ....................... 6.3 Posing and solving problems as a remediation of Einstellung effect ................................................................. 6.4 Posing problems for water jar experiments using a spreadsheet ........................................................................ 6.5 Einstellung effect in finding areas on a geoboard .................

95 99 102

105 105 107 110 112 112 113 113 114 115 116 116 119 123 125 128 130

133 133 138 141 143 144

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6.6 Einstellung effect in solving algebraic equations and inequalities ............................................................................ 6.7 Einstellung effect in solving trigonometric inequalities ....... 6.8 Using technology to pose problems that might lead to Einstellung effect ................................................................. 6.9 Einstellung effect in solving logarithmic inequalities ........... 6.9.1 Simultaneous extension and contraction of solution set.................................................................. 6.9.2 Extension of solution set ............................................ 6.10 Solving logarithmic inequality (6.21) in the general case..... 6.10.1 The case n = 2k ........................................................... 6.10.2 The case n = 2k + 1..................................................... 6.11 Summary ...............................................................................

149 153 156 159 159 162 166 166 172 176

Chapter 7 Explorations with Integer Sequences as TITE Problem Posing ........... 179 7.1 Introduction ........................................................................... 179 7.2 Exploring patterns formed by the last digits of the sums of powers of integers ................................................................. 180 7.3 Discovering patters in the last digits of the polygonal numbers ................................................................ 186 7.3.1 The triangular number sieves ..................................... 186 7.3.2 Triangular number sieves and the last digits of their terms .................................................................. 188 7.3.3 Rises and falls in permutations ................................... 189 7.3.4 Connecting triangular and square numbers within the multiplication table ..................................................... 189 7.3.5 The square number sieves .......................................... 191 7.3.6 The pentagonal number sieves ................................... 193 7.3.7 The general case of the m-gonal number sieves ......... 195 7.4 Patterns in the behavior of the greatest common divisors of two polygonal numbers..................................................... 200 7.5 Exploring sequences formed by the sums of powers of integers .................................................................................. 202 7.6 Exploring sieves developed from the sums of powers of integers .................................................................................. 204

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7.7 Summary .............................................................................. 207 Appendix ............................................................................................. 8.1 Spreadsheets included in Chapter 1 ...................................... 8.2 Spreadsheets included in Chapter 2 ...................................... 8.3 Spreadsheets included in Chapter 3 ...................................... 8.4 Spreadsheets included in Chapter 4 ...................................... 8.5 Spreadsheets included in Chapter 6 ...................................... 8.6 Spreadsheets included in Chapter 7 ......................................

209 209 210 211 212 212 213

Bibliography

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Index

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Chapter 1

On the Genesis of Problem Posing in Mathematics 1.1 Problems from the first printed arithmetic Problem posing has been used in mathematics education as a teaching method for quite a long time. It can be traced back to the 15th century when the first printed book on arithmetic, written (by an unknown author) for those students “who look[ed] foreword to mercantile pursuits” [Smith, 1959, p. 1], included real-life problems posed to explain how the rules of arithmetic work in trade. Connection of arithmetic to real life was then a new idea in mathematics education. In this bygone book, the rules of arithmetic were put in context in which numbers and operations on numbers could be used to describe concrete things and actions on them as related to trade. This is what the 21st century educators have been trying to do when explaining a physical meaning of the rules of multiplying and dividing fractions. It is through posing problems by our educational forefathers, who appreciated the applied nature of mathematics as a field of study, that the subject matter gained its applied flavor, although it is not clear whether they expected the meaning of arithmetical operations to stem from applications. According to Smith [1924], the problems posed in the 15th century book satisfied two major requirements: 1) a problem must be as much concrete as possible (e.g., in a problem about trade, assign names to merchants; a problem about travel includes references to well-known places and individuals); 2) a problem should have students practicing specific rules of arithmetic (e.g., to use the so-called rule of two things by multiplying and adding two given numbers and then dividing the product by the sum). It should be noted that the emphasis on the formal use of such rules of arithmetic, as the rule of two things, points at the roots of teaching mathematics without understanding, primarily through drill and practice. In North America, until the second decade of the 19th century, the

1

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textbooks almost exclusively used the rule method grounded in pure memorization followed by drill and practice [Michalowicz and Howard, 2003]. This is what the modern-day mathematical teaching standards around the world have been trying to change. For one, the National Council of Teachers of Mathematics [2000], the major professional organization of mathematics educators in North America, has been of the strong opinion that students should be offered “opportunities to learn important mathematical concepts and procedures with understanding” (p. 3). Likewise, more than half a century ago, at the very outset of mathematics education reform, progressive mathematics teachers in England advised “that teaching which tries to simplify learning by emphasizing the mastery of small isolated steps does not help children, but put barriers in their way” [Association of Teachers of Mathematics, 1967, p. 3]. Indeed, many applied problems published in the first book on arithmetic and reproduced by Smith [1924] would be considered quite challenging for a 21st century student if taught in a purely formal way, that is, through the use of rules one has to memorize but not necessarily understand. That is why, in what follows, a problem from the past will be used as an example to discuss modern methods of teaching mathematics with emphasis on the use of pictures, diagrams, and technology [e.g., Association of Mathematics Teacher Educators, 2017; Common Core State Standards, 2010; Conference Board of the Mathematical Sciences, 2012; Ministry of Education Singapore, 2012]. Consider a travel problem as presented in [Smith, 1924, p. 330]. Problem 1.1. “The Holy Father sent a courier from Rome to Venice, commanding him that he should reach Venice in 7 days. And the most illustrious Signora of Venice also sent another courier to Rome, who should reach Rome in 9 days. And from Rome to Venice is 250 miles. It happened that by order of these lords the couriers started on their journeys at the same time. It is required to find in how many days they will meet.” Several questions have to be answered in connection with this problem. Question 1. How can one solve this problem using the modern pedagogy of teaching arithmetic?

On the Genesis of Problem Posing in Mathematics

3

Question 2. How can one pose similar problems using conceptual understanding of the problem situation? Question 3. How can one use technology to pose similar problems with friendly solutions? 1.1.1 Solving a 15th century problem using the modern-day pedagogy To answer the first question, let us assume (for simplicity) that the distance from Rome to Venice is the unit distance. While a reference to 250 miles adds concreteness to the problem, the use of this number does not affect the way its solution is structured. Also, one has to assume that both couriers walk with constant speed. In fact, this tacit assumption is true for all pre-calculus travel problems. Then the courier going to Venice will cover 1/7 of the distance in one day. Likewise, the courier going to Rome will cover 1/9 of the distance in one day. Here one can recognize the first appearance of fractions as a result of an operation on whole numbers. Note that 1/7 and 1/9 are not just unit fractions, they are rudiments of the concept of uniform movement establishing relation among distance, time, and velocity. Similar reciprocals of integers appear through solving the so-called work problems where, under certain assumptions, one’s capability of doing work can be interpreted as speed with which one moves. In order to facilitate students’ understanding of fractions, different visual methods of teaching are commonly used nowadays. With this in mind, through a two-dimensional model, used when two fractions are involved (1/7 and 1/9 in the case of Problem 1.1), the whole distance can be represented as a 7x9 rectangular grid with 63 identical cells (Fig. 1.1). To this end, the rectangle is divided both horizontally and vertically in seven and nine parts, respectively. Through the physical meaning of the two fractions, the very grid obtains its physical meaning as well. One can see that the faster courier can cover the whole distance (‘going’ from top to bottom) in seven days and the slower courier (‘going’ from left to right) can cover the whole distance in nine days. When moving towards each other within the grid (the virtual distance from Rome to Venice), the couriers cover 16/63 ( of the grid in one day. At the conclusion of the fourth day, they would have

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covered the entire distance plus 1/63 of it (because the slower courier is short of one cell on the fourth day – the six non-shaded cells of the grid represent the distance left to the slower courier to cover on the fourth day). That is, 4 ∙ 1 . This means that they need less than four full days in order to meet. How can one represent an extra cell as a fraction of a day? There are 16 cells representing one day; therefore, one cell represents 1/16 of a day. Subtracting 1/16 from 4 yields an answer to the problem: 4 3 (days). Note that 3 , which is the reciprocal of the fraction

– the distance covered by two couriers in one

day. This means that the time till they meet is the reciprocal of the distance they cover in one day. In order to better understand the meaning of the last statement, consider a simpler situation, when 1/2 of a distance is covered in one day. It is quite obvious that the whole distance will be covered in two days. Note that the number 2 (time) is the reciprocal of the number 1/2 (velocity) in our rudimental understanding of uniform movement. In the case of a non-unit fraction, one can use a one-dimensional model for fractions (Fig. 1.2) to show how to move from one unit of measurement to another. In order to find what fraction of 16/63 (the measure of one day in terms of distance) is 15/63 (the far-right segment in Fig. 1.2), one has to divide the latter fraction by the former one to get 15/16 (of a day). Once again, the answer is 3 (days). Alternatively, the answer can be given as 3 days, 22 hours and 30 minutes after calculating 15/16 of 24 in terms of hours and minutes.

Fig. 1.1. Solving the distance problem using rectangular grid.

On the Genesis of Problem Posing in Mathematics

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Fig. 1.2. The sum of the four fractions is equal to one.

1.1.2 Posing 15th century-like problems through conceptualization Now we can answer the second question: how can one use our understanding of the problem situation in order to pose similar problems so that the number of days is an integer? Put another way, by asking this question, one poses the problem: When is the reciprocal of the ratio of the product of two integers to their sum an integer as well? This problem cannot be solved in a purely numeric milieu, without recourse to algebra. , in which, in Algebraically, one needs to explore the equation the context of Problem 1.1, x and y are two integers (things) involved in “the rule of two things” and n is the (unknown) time the couriers need in order to meet somewhere between Rome and Venice. The last equation is whence equivalent to the equation .

(1.1)

For example, when n = 4 one of the solutions of (1.1) is x = 6, y = 12. One can construct the 6x12 grid, and using it as one whole, to see that it is comprised of four groups of 1/12 and four groups of 1/6. Equation (1.1) has quite a long history going back to Babylonian mathematics (about 2000 B.C.). Indeed, when n = 2 equation (1.1) is equivalent to the following relation between “the two things”, xy = 2(x + y), and Babylonians knew how to find x and y [Van der Waerden, 1961]. Furthermore, the case n = 2 relates to finding rectangles with perimeter equal, numerically, to area, a problem known to Pythagoras1. For example, if x and y are integers which measure the side lengths of such a rectangle, then xy=2(x + y). Van der Waerden’s [1961] citation of Plutarch2 is worth mentioning in connection with this unexpectedly 1 Pythagoras 2

(circa 570–495 BC) – a Greek mathematician and philosopher. Plutarch (46 – 120 AD) – a Greek historian of science.

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discovered situation: “The Pythagoreans also have a horror for the number 17. For 17 lies halfway between 16 … and 18 … these two being the only two numbers representing areas for which the perimeter (of the rectangle) equals the area” (p. 96). In other words, the equation xy=2(x + y) has only two solutions, x = y = 4 and x = 3, y = 6, representing 4x4 and 3x6 rectangles. Once again, each rectangle can be turned into a grid with the first one being interpreted as the case of two equal couriers covering the distance between Rome and Venice in four days; the second one being interpreted as the case of unequal couriers covering the distance in three and six days, respectively (each pair of couriers meeting after two days). 1.1.3 Using technology for posing 15th century-like problems The third question deals directly with the main topic of this book: How can one use technology, given the value of n, to solve equation (1.1) in order to pose new distance problems with friendly solutions? To answer this question, note that whereas solving a travel problem one moves from problem’s data to answer, when posing such a problem with technology one moves from (known) answer to (unknown) data. Different technology tools can be used to numerically model equation (1.1). For one, Wolfram Alpha (a powerful computational knowledge engine available free on-line at www.wolframalpha.com) can be used. Fig. 1.3 shows five integer solutions to equation (1.1) generated by Wolfram Alpha in the case n = 6. That is, if in Problem 1.1 the two numbers, 7 and 9, would be replaced by any of the pairs (7, 42), (8, 24), (9, 18), (10, 15), (12, 12), the answer, six days, would be the same in all the five cases. Isn’t it interesting and, thereby, doesn’t it motivate a request for explanation? Students may be asked to practice solving such problems by using a rectangular grid like the one shown in Fig. 1.1. At the same time, using Wolfram Alpha allows for posing a travel problem when the number of days sought is an improper fraction, i.e., . Fig 1.4 shows that when one of the couriers needs 6 days and another one needs 10 days to cover the entire distance, then they meet in 3 days. To formulate such a problem without the use of technology would be quite time consuming.

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Remark 1.1. One can try solving equation (1.1) for other integer values of n and see how the number of solutions depends on n. This experimentation using Wolfram Alpha can motivate interest in requesting explanation as to why, whereas 7 > 6, the case n = 6 yields five solutions and the case n = 7 yields only two solutions. In the following chapters of the book the discussion of questions requesting explanation of computationally obtained information will be shown as having the major influence on the quality of problem posing.

Fig. 1.3. Five whole number solutions to (1.1) with n = 6.

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Fig. 1.4. Three whole number solutions to (1.1) with n = 15/4.

1.2 From a classic problem to using the modern spreadsheet 1.2.1 The birth of the probability theory through problem posing In the 16th-17th centuries, many mathematical problems were posed by educated non-mathematicians in the context of different games, some of them games of chance, which, at that time, had been considered important in the life of society. One such problem, the solution of which (discussed below) is credited to Pascal3, is associated with the birth of the theory of probability: How can one divide stakes in a game between two equal partners that was cut short when the scores were not equal? Obviously, when the scores are equal, then the prize money should be equally divided between two players.

3

Blaise Pascal (1623–1662) – a French mathematician and physicist.

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The problem appeared in one of the first printed books on mathematics by Pacioli [1494] where the solution was to divide the stakes in the ratio of the achieved scores. For example, if the players achieved the score 1:3 by the time the contest was cut short, $20 prize money has to be divided into $5 and $15. The problem can also be found “in Italian mathematical manuscripts as early as 1380” [Ore, 1960, p. 414]. Pacioli’s treatment of the problem was criticized by Cardano4, who, however, also proposed an unacceptable solution [Ore, 1953]. Only Pascal’s solution, which “may well be considered a decisive breakthrough in the history of probability theory” [Ore, 1960, p. 413], was accepted by mathematicians. To clarify, consider the problem in the following form by using names of the players just as an unknown Italian author of the 15th century arithmetic book did to make problems more concrete [Smith, 1924]; in other words, to make it pedagogically coherent (Chapter 2). Problem 1.2. Suppose that two equal partners, Ron and Tom, play chess until one of them scores five wins and thus receives the entire stake, say, $32. For some reason, the game was cut short at the score two (Ron) to three (Tom). In that case, the prize money has to be divided. Because of unequal scores, the division may not be in equal parts. How can this be done fairly? Pascal suggested to consider the unfinished part of the contest and to divide the prize not in the ratio of two to three (the achieved scores) but proportionally to the probabilities of winning the contest by each player had it not been cut short. First, in such an imaginary situation, one has to determine the exact number of games that have to be played to allow one of the two players to win. In order for Ron to win, he has to win at least three games while allowing Tom to win at most one game. So, the exact number of games they have to play in an imaginary situation is four. In order to solve the problem, Pascal considered a single probability space with four games. This implies that all four games must be played even if Ron scores five wins after playing three games in order to allow Tom to win one game. Likewise, even if Tom scores five wins after playing two 4

Girolamo Cardano (1501–1576) – an Italian mathematician and physician.

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games, all four games must be played in order to allow Ron to win two games. Such position by Pascal is described by Todhunter [1949] in the following words: “although it is quite possible that the game may be finished in two trials or three trials, yet we are at the liberty to conceive that the players agree to have four trials, because, even if the game be decided in fewer than four trials, no difference will be made in the decision by the superfluous trial or trials. Pascal puts this point very clearly” (pp. 13-14, italics in the original).

Fig. 1.5. The first six rows of Pascal’s triangle.

In order to solve Problem 1.2 following the idea of Pascal, note that each such game between two equal partners is equivalent to tossing a fair coin – an experiment with exactly two outcomes, with the probability 1/2 both for head (H) and tail (T). It is known from the history of mathematics (e.g., [Kline, 1985]), that Pascal used his famous triangle (Fig. 1.5), nowadays called Pascal’s triangle (although known long before the 17th century), for recording sample spaces of an experiment of tossing a coin one, two, three, four, etc. times. For example, when a coin is tossed four times, the following combinations of heads and tails are possible: {HHHH}, {HHHT, HHTH, HTHH, THHH}, {HHTT, HTHT, HTTH, THTH, THHT, TTHH}, {HTTT, THTT, TTHT, TTTH}, and {TTTT}; these five sets having cardinalities represented, respectively, through the string of numbers 1 4 6 4 1. This string is the fifth line of Pascal’s triangle that corresponds to four tosses of a coin. In the case of the event with exactly three heads (with head

On the Genesis of Problem Posing in Mathematics

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meaning success and tail meaning failure) out of four tosses (playing four games) there are four possible independent outcomes (as the above second set indicates), each having the probability equal to the product . Therefore, the probability of winning exactly three games out of four is equal to 4 ∙

.

However, according to Pascal, there exists another winning possibility for Ron – to win all four games (alternatively, the coin turns up head four times), a single outcome (as the above first set indicates) . Because winning three games out of four with the probability and winning all four games are two incompatible events, the corresponding probabilities have to be added; thus, the probability that Ron wins in the imaginary situation is equal to 1/4 + 1/16 = 5/16. Consequently, if the game were not cut short, Tom has the probability to win equal to 11/16. This shows that a fair division of prize money has to be in the ratio 5:11; in the case of $32 prize, the sums of monies given to each player have to be $10 and $22. Note that solution by Pacioli [1494] would give $12.80 to Ron and $19.20 to Tom as 2/5 and 3/5 of $32, respectively. While the difference in moneys between the two approaches to the problem is not significant, the role of Pascal in understanding how probabilistic reasoning can be used in a problem-solving situation is of the monumental historical proportion. 1.2.2 Using a spreadsheet as a problem-posing tool Whereas the problem about division of stakes has been discussed in the literature from a number of perspectives [Borovcnik and Kapadia, 2014; Chernoff and Zazkis, 2011; Edwards, 1982a], the use of technology in modeling the problem appeared only recently [Abramovich and Nikitin, 2017]. Why was technology suggested to be used in the context of the problem? The reason is the same as in the case of the travel problem – to advise and facilitate posing similar problems but with different data. How can one alter data associated with Problem 1.2 and, in doing so, to pose its different variations? Without using technology, verification of the solvability of a problem in an educational context could be quite cumbersome and time consuming for a teacher who wants to offer a

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multitude of problems on the division of stakes to their students. At the same time, an appropriately programmed electronic spreadsheet, a tool commonly available in the modern-day classroom, can be used to pose such problems. This is especially beneficial for high school teachers who can discuss this problem in class and then give students multiple problems, which are numerically coherent (Chapter 2), to solve as an application of Pascal’s triangle to a classic problem about the division of stakes. The spreadsheet shown in Fig. 1.6 (see Appendix for programming details) is designed to model this problem in the general case of N games, with n to m ratio of scores when the competition of two players with the probabilities to win a game equal p and q, respectively, was cut short. Instead of using Pascal’s triangle, something that is only possible when p = q = 1/2, Bernoulli’s formula5 1 , which computes the binomial probability of obtaining exactly r successes in n trials with the probabilities of success and failure equal to p and 1 – p, respectively, was used to program the spreadsheet. This formula can be found in many textbooks on introductory probability theory (e.g., [Feller, 1968]) and it also is a part of advanced high school mathematics curriculum (e.g., [Dai, 2002; Posamentier, Smith, and Stepelman, 2006]). In this general case, the number of virtual games to be played is equal to (N – n) + (N – m – 1) = 2N – (m + n + 1). For example, when a five-game (N = 5) contest was cut short at the score 2 (n = 2) to 3 (m = 3), the difference 5 – 2 = 3 represents the number of virtual games Ron has to play in order to win and the value 5 – 3 – 1 = 1 represents the number of virtual games Tom can still win without winning the imaginary contest. The spreadsheet of Fig. 1.6 shows the case N = 10, n = 3, m = 7, p = q = 1/2; that is, when a 10-game contest between two equal players is cut short at the score three to seven, the stakes have to be divided in the ratio of 23/256 to 233/256. The denominator 256 points at the number that may be chosen to represent the amount of prize money in a new problem. For example, if the stake is $64, the sums of monies given to each player would be $5.75 (cell J3) and $58.25 (cell J4). Other combinations of N, 5

Jacob Bernoulli (1654–1705) – a Swiss mathematician, a notable representative of the famous family of mathematicians.

On the Genesis of Problem Posing in Mathematics

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n, and m can be tried to generate numerically coherent (Chapter 2) problems about the division of stakes. Furthermore, the spreadsheet can calculate solution proposed in the 15th century book [Pacioli, 1494] when the stakes were divided in the ratio of the achieved scores by the time the contest was cut short. These sums are $19.20 (cell K3) and $44.80 (cell K4). One can see that the larger the stakes, the larger is the difference that Pascal’s and Pacioli’s solutions yield. 1.2.3 Duality of the spreadsheet’s use The proposed spreadsheet-based computational environments may not only be used to formulate new solvable problems. One of the main pedagogic benefits computers play in mathematics teaching and learning is the enhancement of conceptual development. With this in mind, one can also be asked to explore the situations for the triples N, n = N – 2, and m = N – 3 to see that for any value of N ≥ 3 we have the same division of stakes. How can this observation be explained conceptually? Without loss of generality6, one can set N = 3 so that all the situations described by the triple (N, N – 2, N – 3) are equivalent to (3, 1, 0). Indeed, the number of virtual games is equal to 2N – (N – 2 + N – 3 +1) = 4 and the player with the score N – 3 either wins three games or all four games. The same situation is described by the triple (3, 1, 0), that is, when a 3-game competition was cut short after one of the players won the first game. Likewise, the gaming situations described by the triple (N, N – k, N – k – l) can be shown being equivalent to the triple (k + l, k, 0). So, the role of the spreadsheet in the context of the problem of the division of stakes is at least twofold: to generate new problems with friendly solutions (especially in the case of mathematics teacher education) and to use numerical evidence as a means of facilitating conjecturing and enabling conceptualization (important for all populations of students).

6

The expression “without loss of generality” is used in mathematics to indicate that the reduction of a general case to a special case simplifies reasoning without affecting the generality of an argument because the special case includes all needed characteristics of the general case.

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Fig. 1.6. The division of stakes spreadsheet.

1.3 The Problem of the Grand Duke of Tuscany As was mentioned above, gambling problems had often been posed by non-mathematicians who wanted to enhance their mathematical erudition towards becoming more informed gamblers. For one, the Grand Duke of Tuscany (Cosimo II de’ Medici), the benefactor of Galileo Galilei7, had asked the latter to explain mathematically the following gambling observation: when rolling three dice, why does the number 11 appear more often than the number 12, while the number 8 does not appear more often than the number 13? Mathematically, this question might have been asked because both 11 and 12 can be partitioned in three unordered positive integer summands not greater than six in exactly six ways. Indeed, 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. Likewise, both 8 and 13 can be partitioned in three unordered positive integer summands not greater than six in exactly five ways. Indeed, 8=1+1+6=1+2+5=1+3+4=2+2+4=2+3+3 and 13 = 1 + 6 + 6 = 2 + 5 + 6 = 3 + 4 + 6 = 3 + 5 + 5 = 4 + 4 + 5. But despite similarities, the pairs of integers (11, 12) and (8, 15), as outcomes of rolling three dice, have differences in the recorded outcomes observed in a long run of experiments. Two things have to be explained. The first one is why 11 appears more often than 12. The second one is why 8 appears as often as 13. 7 Galileo Galilei (1564-1642) –the father of all major scientific developments in the 17th century Italy.

On the Genesis of Problem Posing in Mathematics

15

Note that a triple of unequal numbers can be permuted in six ways, a triple in which only two numbers are the same can be permuted in three ways, and a triple of equal numbers does not allow for any distinct permutations. Analyzing the above partitions of 11 and 12 in three summands, one can see that in the case of 11 there are three triples of the first kind, three triples of the second kind, and no triple of the third kind. At the same time, in the case of 12 there are three triples of the first kind, two triples of the second kind, and one triple of the third kind. So, in the case of 11 simple arithmetic yields 27 (= 3 ∙ 6 3 ∙ 3 possible outcomes when three dice are rolled. In the case of 12, because of the presence of the triple (4, 4, 4) the number of possible outcomes is equal to 25 (= 3 ∙ 6 2 ∙ 3 1 ∙ 1 . Analyzing the above partitions of 8 and 13 we have 21 (= 2 ∙ 6 3 ∙ 3 possible outcomes for each number. This explains why 11 was observed appearing more often than 12 and 8 was observed appearing as often as 13.

Fig. 1.7. Six unordered partitions of 11.

One can use a spreadsheet to numerically model the outcomes of rolling three dice by displaying both unordered and ordered partitions of 11 (Figs 1.7 and 1.9) and 12 (Figs 1.8 and 1.10). Similarly, the cases of 8 and 13 can be confirmed computationally. To clarify the meaning of numbers in the tables of Figs 1.7 – 1.10, note that the triple (2, 3, 6) in Fig. 1.7 (cells B4, E2, E4, respectively) represents the case 11 = 2 + 3 + 6 and the triple (2, 6, 3) in Fig. 1.9 (cells B4, H2, H4, respectively) represents the case 11 = 2 + 6 + 3. A scroll bar attached to cell B2 makes it possible to explore all cases in the range [3, 18], where 3 = 1 + 1 + 1

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and 18 = 6 + 6 + 6 (Fig. 1.11). The programming details of the spreadsheets are included in Appendix. Consequently, a variety of questions can be posed using these tables.

Fig. 1.8. Six unordered partitions of 12.

For example, one can be asked to find whether a sum different from 11 has the same chances to appear on the faces of three dice. By answering this question with the help of the spreadsheet, one can realize that the numbers in the range [3, 18] can be put in pairs each element of which has equal chances to appear as a sum on the faces of three dice (see Fig. 1.11). In all, according to the rule of product8, there are 216 (= 63) differently ordered sums of three positive integer summands not greater than six. In other words, the number 216 is the cardinality of the sample space of rolling three dice and recording the sum of spots on their faces. One has to appreciate observational skills of the Grand Duke of Tuscany as the difference between the probabilities of the appearances of 11 and 12 is less than 1%. Indeed, 27/216 – 25/216 = 1/108 < 1%. Likewise, one can “roll” (using a spreadsheet) three dice and record the product of spots on their faces to see the product 12 being recorded 15 8 In mathematics, the rule of product means the following: if object A can be selected in m ways and if, following the selection of A, object B can be selected in n ways, then the ordered pair (A, B) of the two objects can be selected in mn ways. For example, if cafeteria offers three types of soft drinks and five types of donuts, then one can select a snack comprised of a soft drink and a donut in 3 5 15 ways.

On the Genesis of Problem Posing in Mathematics

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times. In all, there are 216 differently ordered products that can be recorded through rolling three dice.

Fig. 1.9. 27 ordered partitions of 11.

Fig. 1.10. 25 ordered partitions of 12.

Fig. 1.11. The table displays the number of outcomes when rolling three dice.

1.4 Conjecturing as posing problems to find proof A French mathematician Pierre de Fermat (1607–1665) is probably best known for his famous discovery (dated 1637) that it is not possible to

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extend the Pythagorean equation beyond the second power. Note that the relation 3 4 5 is one of infinitely many solutions to the Pythagorean equation. Quite a few such solutions can be found by typing “Pythagorean triples” in the input box of Wolfram Alpha. But already when n = 3, the equation does not have integer solutions. This discovery, nowadays known as Fermat’s Last Theorem (FLT), remained a conjecture for some 350 years until it was proved by an English-American mathematician Andrew Wiles [Wiles, 1995]. Fermat himself proved it for n = 4 by the so-called method of infinite descent [Edwards, 1977] and great Euler9 adopted Fermat’s method of proof to the case n = 3. In addition, in 1788, Euler posed the following problem (that even for Euler was a conjecture): Prove that it is not possible to represent a perfect n-th power through the sum of fewer than n like powers. The importance of this conjecture is that, if true, it would have implied the correctness of FLT. Indeed, when n > 2, the representation of through the sum of two like powers, , would not be possible. In the case of Euler’s conjecture, it took almost two centuries to find a counter-example (in 1966, via computer search): 144 27 84 110 133 [Stark, 1987, p. 146]. A more complicated counter-example, but for the case n = 4 was found in 1988: 20,615,673 2,682,440 15,365,639 18, 796,760 [Singh, 1997, p. 159]. One can use Wolfram Alpha to verify the correctness of the two counter-examples (representing a fourth power as a sum of three like powers and a fifth power as a sum of four like powers). However, to find those equalities required not only the availability of powerful computers, but highly specialized knowledge of computational mathematics. Note that even in the absence of a counter-example to the Euler’s conjecture, the latter may not have been considered as a proof of FLT. Rather, the conjecture was formulated as a problem to be solved either in affirmative or in the negative. In the former case, it would have proved FLT. In the latter case (requiring just a counter-example), the conjecture would have demonstrated a fallacious approach to proving FLT. 9

Leonhard Euler (1707–1783) – a Swiss mathematician, the father of all modern mathematics.

On the Genesis of Problem Posing in Mathematics

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There are many conjectures that remain to be either proved or disproved. Some of them, just as FLT, have a simple formulation and can be mentioned to conclude this section. One such conjecture stems from the observation that even numbers (greater than 2) can be represented as a sum of two prime numbers (perhaps, in more than one way). Consider the first nine prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23. Using these primes, one can represent many even numbers in that way; for example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, 30 = 7 + 23 = 11 + 19 = 13 + 17. This observation is known as the Goldbach10 conjecture (neither proved nor disproved as of the time of writing this book): every even number greater than two is a sum of two prime numbers. Another simply formulated conjecture also deals with prime numbers. While already Euclid11 proved that there are infinitely many prime numbers, the twin prime conjecture states that there are infinitely many prime numbers of the form p + 2 where p is a prime number. In other words, there are infinitely many prime numbers that differ by the number 2. For example, among the primes listed above we have 5 3 2, 7 5 2, 13 11 2, 19 17 2. Note that except the first pair of twin primes, the other three pairs are separated by a multiple of six: 6, 12, and 18. One can immediately find a counter-example to a possible generalization (conjecture) that every multiple of six separates twin primes: e.g., 24 4 ∙ 6, yet 25 is not a prime number. 1.5 Problem posing in a classic context as a springboard into experimental mathematics A historically important source of problems to be solved is presented in the journal “Mathematical Questions and Solutions from Educational Times” published in London, England, in the 19th-20th centuries. Often, this journal published mathematical questions formulated in the form of requesting a proof of a certain proposition. Consider the case of 10 11

Christian Goldbach (1690–1764) – a German mathematician.

Euclid – the most prominent Greek mathematician of the 3rd century B.C. The terms Euclidean plane and Euclidean algorithm mentioned in the following chapters of this book are named after Euclid.

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triangular numbers – perhaps, one of the most commonly known subsets of natural numbers. Indeed, triangular numbers emerge when one adds consecutive natural numbers starting from one. There are many real-life situations that motivate such addition; e.g., counting handshakes among a group of people, counting the number of ways two books can be selected from a bookshelf, counting the number of billiard balls put together in an equilateral triangle frame. Consequently, many problems/questions involving triangular numbers can be formulated and literature sources in which such problems can be found are in abundance. 1.5.1 Triangular numbers with identical digits For example, one question of that kind was posed by Youngman [1905] in the above-mentioned journal who asked for a proof that the number 666 is the largest triangular number comprised of the same digits. Two proofs were presented in response. Using several concepts of number theory (that are beyond the scope of this book), it was proved that in addition to the obvious three one-digit numbers 1, 3, and 6, there are only three more numbers – 55, 66, and 666 – among all integers with at most 30 digits. Note that limiting proof to integers even with such a large number of digits may not be accepted as a formal proof – after all, as was mentioned above, the absence of a counter-example to the Euler’s conjecture, something that might lead one to believe in its correctness, is not suffice to be considered as proof of FLT. Likewise, the absence of a triangular number with more than 30 identical digits does not prove that 666 is the largest triangular number comprised of the same digits. 1.5.2 Triangular number sieves The context of triangular numbers and their consideration in terms of digits prompts another problem to be posed. To this end, one can note that the number 6 is the third triangular number and the second hexagonal number. In general, every other triangular number is a hexagonal number; that is, every triangular number of rank 2n – 1 is a hexagonal number of rank n. Indeed, the sequences 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, and 1, 6, 15, 28, 45, 66, 91, 120

On the Genesis of Problem Posing in Mathematics

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represent, respectively, the first 15 triangular numbers and the first eight hexagonal numbers. That is, the number 120 is both the triangular number of rank 15 and the hexagonal number of rank eight. In particular, 55, 66, and 666 are triangular numbers of ranks 10, 11, and 36, respectively; therefore, only 66 is a hexagonal number. One can see that by eliminating triangular numbers of even ranks, the sequence of hexagonal numbers results. Consequently, one can refer to hexagonal numbers as the triangular number sieve of order one. For example, as mentioned in [Abramovich and Leonov, 2009], the sequence of numbers 1, 2, 5, 13, 34, 89, …, comprised of every other Fibonacci number and described by the difference equation 3 , 1, 2, is called the Fibonacci sieve of order one. This difference equation was used as one of the tools in proving the famous Hilbert’s Tenth Problem (Matijasevic, 1971), one of the 23 problems posed as a research program for the 20th century by a German mathematician David Hilbert (1862– 1943) in his 1990 address to the International Congress of Mathematicians (see the next section for more information). Just as the sieve of Eratosthenes (see footnote 15 in Chapter 2) eliminates composite numbers from the set of natural numbers, the triangular number sieve of order one eliminates every second triangular number. But unlike the latter sieve, which can be described by the quadratic function , 2 1 , the sieve of Eratosthenes does not have formal mathematical model that generates prime (or eliminates composite) numbers. While there are infinitely many prime numbers, no formula is known to produce a prime number of arbitrary rank. In the fashion of experimental mathematics [Arnold, 2015; Borwein and Bailey, 2004], one can continue eliminating every other number from the triangular number sieve of order one to get the triangular number sieve of order two, , , then do the same with the sequence , to get , , and so on. A problem to be posed is to find a formula for the triangular number sieve of order k, that is, , . A solution to this problem will be presented in Chapter 7 where explorations of that kind with triangular and other polygonal numbers will be carried out.

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1.6 Problem posing as setting up a research program Sometimes, mathematical problem posing aims at charting the program of research for the years to come. The most famous case of that kind is David Hilbert’s keynote address to the 1900 International Congress of Mathematicians in which he posed (presented) to the mathematical community 23 problems from different areas of mathematics including logic, number theory, geometry, algebra, probability theory, differential equations, and calculus of variations. All the problems are quite difficult and are beyond the scope of this book. But Hilbert’s views about problem posing and solving, about the genesis of mathematical ideas and their development presented in his address [Hilbert, 1902] are extremely important for our understanding of the nature of mathematical problem posing and solving. Following are several quotes from the address that would be referred to throughout the book as appropriate. Among general characteristics of a good mathematical problem that were mentioned in the address are “clearness and ease of comprehension … for what is clear and easily comprehended attracts, the complicated repels us” [ibid, p. 438]. At the same time, “a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts” [ibid, p. 438]. As will be shown in Chapter 2, the notion of pedagogical coherence of a problem, especially when the ease of posing a problem with the help of technology may instantly turn problem solving into an unintended derision of students’ exuberance, stems from such an impeccable perspective on mathematical problem posing and solving. Undoubtedly, Hilbert’s perspective on what it means to pose a good problem is pertinent to any grade level of mathematics education. As we already saw in the previous sections of this chapter, mathematical problems are reflections on activities that have been considered significant at a certain time in the history of civilization. Indeed, as Hilbert [1902] put it, “Surely the first and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena. … But in the future development of a branch of mathematics, the human mind, encouraged by the success of its solutions,

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becomes conscious of its independence. It evolves from itself alone, often without appreciable influence from without, by means of logical combination, generalization, specialization, by separating and collecting ideas in fortunate ways, new and fruitful problems, and appears then itself as the real questioner. … In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new branches of mathematics, and while we seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus at the same time advance most successfully the old theories” (p. 440). For example, the problem of the division of stakes (section 1.2.1) was the first instance of using probabilistic reasoning in problem solving. From Pascal’s revolutionary breakthrough in finding the proper way of dividing stakes, the probability theory was born. Then, the interpretation of the entries of Pascal’s triangle as the binomial coefficients allowed for the extension of Pascal’s recording of coin tossing to Bernoulli’s formula and to the Law of Large Numbers, something that have been used ever since as the basic tools of the theory. In turn, the Law of Large Numbers allowed for the experimental approximation of the value of π by dropping a small needle either on a board partitioned by a series of equidistant parallel lines (Buffon’s12 needle problem) or on the grid paper (in the words of Laplace13 [1812, p. 461], on “a floor being divided into small square tiles”) as the fall of the needle (the length of which is smaller than the distance between any two lines) depends on its angle of inclination with regard to the lines so that the number π comes into play (e.g., [Uspensky, 1937]). This experimental approach to the approximation of π not only offered a new (though not necessarily better) solution to an old problem but it also gave birth, although much later (in the mid-20th century), to the Monte Carlo method [Metropolis and Ulam, 12

Georges-Louis Leclerc Buffon (1707–1786) – a French naturalist and mathematician. Pierre Simone Laplace (1749–1827) – a French mathematician and scientist, one of the greatest scholars of all time. 13

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1949] in which randomness is used to solve essentially deterministic problems through a statistical approach. The modern-day students who are advised to “make sense of problems and persevere in solving them” [Common Core State Standards, 2010, p. 6], would benefit from knowing that “every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts” [Hilbert, 1902, p. 444]. So was the case with both Fermat’s Last Theorem and the Euler’s conjecture mentioned above in section 1.4. Whereas the former, as a problem inadvertently posed by Fermat for others to prove, was eventually solved in the affirmative (thus having the word theorem in it name), the latter was solved in the negative using computer search and thus it remains in the history of mathematics as a disproved conjecture. Posing problems for students to solve that provide both positive and negative experiences, is extremely important. As a simple example, a young learner of mathematics can be asked to represent both 9 and 8 (in that order) as the sum of consecutive natural numbers. Whereas 9 = 4 + 5 = 2 + 3 + 4, the number 8 does not have the representation sought. Regarding the “use of appropriate tools strategically” [Common Core State Standards, 2010, p. 7], one should have in mind that “The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas; and no mathematician could spare these graphic formulas, any more than in calculation the insertion and removal of parentheses or the use of other analytical signs” [Hilbert, 1902, p. 443] and that “new signs … we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts” [ibid, p. 442]. For example, a spreadsheet pictured in Fig. 1.7 is an example of a new sign of the digital era that represents kind of a diagram or graph which remind us of the experiment when rolling three dice allows for six possibilities to get eleven. Likewise, the rectangular grid pictured in Fig. 1.1 represents a graphic formula which solves the distance problem. Finally, as Hilbert put it, “… it is an error to believe that rigor in the proof is the enemy of simplicity” [ibid, p, 441]. Indeed, what made the

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problem by the Grand Duke of Tuscany (section 1.3) difficult for nonmathematicians was the lack of rigor in representing the number 11 as a sum of three natural numbers. It is rigor that requires using the commutative property of addition. The appreciation of this apparently hidden property eliminated difficulty in relating experiment to written diagrams involving the summation of the spots on the three faces. As will be shown in Chapter 6 of this book, the absence of rigor often results in the so-called Einstellung effect in problem solving [Luchins, 1942]. 1.7 Summary In this chapter, several classic problems were selected from rich mathematical heritage of humankind to demonstrate their didactic value in the modern-day classroom and to show how conceptualizing these problems in terms of contemporary pedagogy allows one to use technology as a tool of posing structurally similar problems. In particular, problems that provided foundation for the development of the theory of probability were considered. Another type of problems with rich mathematical and computational history that were considered in this chapter dealt with the tradition, going back to Pierre de Fermat, of formulating conjectures so that rigorously proving them or defying through a counter-example has long been considered by mathematicians as a problem to solve. In other words, whereas computations can be used to find a counter-example defying a conjecture, a pure mathematical thought is necessary for an affirmative outcome. Digital support of the ideas discussed in the chapter included the use of Wolfram Alpha and a spreadsheet. The chapter also highlighted a historical aspect of problem posing by presenting David Hilbert’s views about the development of mathematics shared in his (famous among mathematicians) 1900 keynote address in which a research program in mathematics for the 20th century was formulated. The next chapter will demonstrate how a theory of mathematical problem posing in the technological paradigm can enrich mathematics pedagogies of a teacher education program at the elementary level.

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Chapter 2

From a Theory of Problem Posing to Classroom Practice of the Digital Era 2.1 Problem posing as educational philosophy At the end of the 19th century, the notable Italian educational reformer Maria Montessori, by inquiring, “Should not the child ... some day brace himself to a real effort, compelling him to carry out a necessary, rather than a chosen, task?” [Montessori, 1917, p. 1], had developed a notion of problems to be posed by pupils, a progressive pedagogical approach that nowadays has exemplified itself as an important characteristic of a student-centered classroom [Jones, 2007]. In such a classroom “children create their own curriculum materials” [Thayer-Bacon, 2013, p. 130] as part of “the idea that the child is not just a smaller version of the adult ... [but rather] children should be free to choose their own work” (ibid, p. 48). Teachers, on their part, as a Japanese philosopher of education of the first part of the 20th century Tsunesaburo Makiguchi argued, should be “channeling student efforts into better ways of formulating ideas for themselves” [Makiguchi, 1989, p. 122] in order to support students’ interest in learning, something that may never be neglected or restrained. Education, like work, should develop pleasure and happiness in an individual. What could be more pleasant for a student than the feel of ownership of ideas that the whole class discusses? In the case of mathematics learning, what could make a student happier than posing a worthwhile problem for the whole class to solve or asking a good question for the whole class to discuss under the guidance of a teacher? These perspectives on teaching and learning by Makiguchi are clearly either motivated by or in agreement with Montessori’s educational philosophy. A pedagogy that, instead of emphasizing memorization of arithmetical tables, approaches mathematics through problem posing and helps children appreciate their teachers’ focus on “experiment, observation, evidence of proof, the recognition of new phenomena, their 27

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reproduction and utilization, undoubtedly place it [mathematics] among the experimental sciences” [Montessori, 1917, pp. 73-74]. An emphasis on experimentation through problem posing, away from memorizing tables and formulas without understanding the meaning of numbers in the tables and letters in the formulas, suggests the importance of mathematical experiment as a teaching method to be integrated across the grades. Mathematically, these ideas have ancient roots as evidenced by the following admission made by Archimedes14 in a letter to Eratosthenes15: “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” [Archimedes, 1912, p. 13]. The advent of computers has elevated the possibilities of “a mechanical method” to the qualitatively new level enabling, in the spirit of Archimedes, to maintain “a proper balance between the role of proof and the role of experiment, including the role of the senses, as tools in the search for new mathematical facts” [Mandelbrot, 1994, p. 97]. At the time when the Italian educator developed her experimental pedagogy based on the philosophical idea that emphasizing children’s “liberty ... will lead to the maximum development of character” [Montessori, 1917, p. 6], an American psychologist William James studied the art of teaching from the point of view of psychology. He argued that the best way to develop students’ interest in a subject matter is to focus teaching on topics that are within their basin of attraction: “Any object not interesting in itself may become interesting through becoming associated with an object in which an interest already exists” [James, 1983, p. 62]. Indeed, one is unlikely to ask a question or pose a problem, which are outside of their interest. The main challenge here is, of course, to create conditions that motivate students to ask questions. As 14

Archimedes (circa 287-212) – the greatest mathematician of antiquity (Greece) and beyond. 15 Eratosthenes – a Greek scholar of the 3rd century B.C., credited with developing the process of identifying prime numbers among natural numbers, known as the sieve of Eratosthenes.

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Mandelbrot [1994], in a plenary lecture on experimental geometry and fractals at the 7th International Congress on Mathematical Education, advised the audience: “Motivate the students by that which is fascinating, and hope that the resulting enthusiasm will create sufficient momentum to move them through that which is no fun but is necessary” (p. 86). In other words, the activity of problem posing, when tailored to individual interests, may blur the distinction between the joy of knowing and the necessity of learning. In a similar way, a Brazilian educator Paulo Freire promoted the problem-posing concept of education as an approach which “does not dichotomize the activity of the teacher-student: she is not “cognitive” at one point and “narrative” at another” [Freire, 2003, p. 80] and argued that “problem-posing education ... [being] revolutionary futurity ... corresponds to the historical nature of humankind ... affirms women and men as beings who transcend themselves ... for whom looking at the past must only be a means of understanding more clearly what and who they are so that they can more wisely build the future” (ibid, p. 84). But looking at the past is congruent with the notion of reflective inquiry introduced by an American educational philosopher John Dewey as a problem-solving method that obscures the difference between knowing and doing and aims at integrating knowledge with experience [Dewey, 1933]. Using this method as a catalyst for a mathematical thought can advance mathematics instruction by approaching problem solving as inquiry [Kilpatrick, 2016], foster students’ metacognitive skills [Yu, 2009], and expand action learning pedagogy, proved to be effective in the contexts of business management and teacher development [Boshyk and Dilworth, 2010; Norton, 2009; Revans, 1982], to mathematics education [Abramovich, Burns, Campbell and Grinshpan, 2016]. 2.2 Problem posing in the modern educational context Over the last three decades, mathematical problem posing appeared in a variety of didactic forms [Brown and Walter, 1990; Singer, Ellerton and Cai, 2015; Felmer, Pehkonen and Kilpatrick, 2016] as a way of providing students with experience in exploring mathematical ideas, investigating self-formulated conjectures, and solving problems relevant to students’ interest and background through the reformulation of those already solved. Reformulation of a mathematical problem can be seen as Freire’s

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[2003, p. 84] “looking at the past” educational philosophy with the goal to understand the nature of the original problem in order to use this understanding to “more wisely build the future” learning of mathematics. This position is in agreement with teaching and learning that encourage reflection on what was already done in order to formulate and then find answers to new queries. In particular, the importance of problem posing as a method of encouraging a critical element of mathematical thought in the form of reflective inquiry was emphasized through the so-called Socratic seminars [Chorsempa and Lapidus, 2009] – a learning model that encourages learners to generate questions. Mathematics begins with posing problems in the context of concrete activities “suggested by the world of external phenomena” [Hilbert, 1902, p. 440] and it develops, using terminology introduced by a Russian educational psychologist Lev Vygotsky whose work greatly influenced views on education in the United States, through the transition from dealing with “the first-order symbols … directly denoting objects or actions … [to] the second-order symbolism which involves the creation of written signs for the spoken symbols of words” [Vygotsky, 1978, p. 115]. Put another way, using pictures, diagrams, and manipulative materials allows students to comprehend the meaning of numbers and variables as the basic entities of mathematics and to appreciate the need for numeric/algebraic relations that are comprised of these entities. Therefore, mathematical problem posing may be considered as a subject-oriented strand of a more general educational philosophy and teaching method. Consistent with the observation that “activities are much more effective than conversations in provoking problems” [Isaacs, S., 1930, p. 4], problem-posing activity utilizes the primary character of the first-order symbols (concrete objects) versus the secondary nature of the symbolic description of those objects and the ways they are arranged. Following pedagogical ideas of the major educational document in the United States at the time of writing this book, by taking advantage of the teacher’s presence in the classroom who is supposedly ‘a more knowledgeable other’, the modern student develops “the ability to decontextualize [from the first-order symbols] and contextualize … in order to probe into the referents for the [second-order] symbols

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involved” [Common Core State Standards, 2010, p. 6, italics in the original]. As an illustration, consider the diagram of Fig. 2.1 in which, using square tiles as the first-order symbols, the same quantity of the tiles is represented in three different ways: as a pile, as the base ten imagery, and as its base eight equivalent. The last two representations of the same quantity of the first-order symbols can help one understand the meaning of the second-order symbolism associated with a base system. In other words, the latter type of symbolism “remind[s] us of the phenomena which were the occasion for the formation of [non-decimal bases as] the new concepts” [Hilbert, 1902, p. 442]. Typically, only base ten system is studied at the primary level. However, according to Vygotsky [1962], “As long as the child operates with the decimal system without having become conscious of it as such, he has not mastered the system but is, on the contrary, bound by it. When he becomes able to view it as a particular instance of the wider concept of a scale of notation, he can operate deliberately with this or any other numerical system” (p. 115). With this in mind, a variety of problems can be posed by a teacher with the goal of developing students’ conceptual understanding of a base system. Such problems may include the requests of forming representations in other bases, splitting the tiles in two or more groups (with or without regard to order), each of which representing a nondecimal base, doing addition in that base in order to verify the correctness of symbolic representations.

Figure 2.1. Three different representations of the same quantity.

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Problem posing that originates within a pile of square tiles may bring about other kinds of representations through the first-order symbols. Conceptually rich representations may be created, perhaps by serendipity, having just very basic conceptual understanding of the role of the first-order symbols in developing mathematical ideas. Once a representation through the first-order symbols is interpreted contextually either by a student or by a teacher, it can be used as a springboard into de-contextualization which may lead to the development of the variety of mathematical concepts expressed through the second-order symbolism of “written diagrams and … graphic formulas” [Hilbert, 1902, p. 443]. 2.3 Learning to ask questions about posed/solved problems In the context of problem posing and problem solving it is important that one distinguishes between two types of questions that can be formulated to become a problem [Isaacs, N, 1930]: questions seeking information and questions requesting explanation of the information obtained. Similar to two types of signs – the first-order symbols and the secondorder symbolism – one can refer to questions seeking information as the first-order questions and those requesting explanation as the secondorder questions. Whereas the first-order questions can be answered using different methods, it appears that not all methods can be used to provide an explanation of what was obtained in search for information, that is, to provide an answer to a second-order question. Often, the request for explanation is an intelligent reflection on a method that provided information. For example, the number of rectangles within a 3x3 grid (Fig. 2.2, left) can be found by referring to Pascal’s triangle (Fig. 1.1) and coin tossing as a way of avoiding the explicit use of binomial coefficients (cf. Pascal’s treatment of Problem 1.2, Chapter 1). To clarify, note that any rectangle is defined by two pairs of parallel sides. Within the 3x3 grid, one pair can be chosen by selecting two sides out of four; that is, in six ways (in terms of Pascal’s triangle and coin tossing, the number 6 being the element of its fifth row, 1 4 6 4 1, describes the cardinality of the sample space {HHTT, HTHT, HTTH, THTH, THHT, TTHH} when four tosses of a coin yield two heads). But another pair of parallel sides can be

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selected in six ways as well. By the rule of product, two pairs of sides that determine a rectangle within the 3x3 grid can be selected in 36 ways. That is, the number of rectangles within a 3x3 grid is equal to 36; among them are 12 + 22 + 32 = 14 squares. Similar to counting squares, one can count cubes within a 3x3x3 cube (Fig. 2.2, right) as the sum 13 + 23 + 33 = 36. Indeed, each of the three layers of the large cube consists of nine unit cubes; each of two pairs of adjacent layers consists of four 2x2 cubes; finally, there is only one 3x3x3 cube. Adding up the number of cubes results in the sum of the first three cubes of natural numbers. This concludes obtaining information about the number of rectangles and the number of cubes.

Fig. 2.2. From seeking information to requesting explanation.

Fig. 2.3. Mapping cubes on gnomons.

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One can recognize that the number of cubes within a 3x3x3 cube is the same as in the number of rectangles within a 3x3 grid. Why is it so? This is a second-order question requesting explanation of information obtained through answering the first-order questions about the number of rectangles and cubes. The explanation can be provided through the recourse to a combination of the first-order symbols and the second-order symbolism. Indeed, all cubes in the 3x3x3 cube can be put in three groups depending on their size. By representing a face of the cube (which is a 3x3 grid) as the 3x3 multiplication table, each such group can be mapped on one of the three gnomons of the table (the gnomons comprised of cells shaded differently in the diagram of Fig. 2.3). Adding up all numbers in the table yields 36. Thus, each cell can be seen as a storage for as many rectangles as the product it represents. The sum of products with the factor three is equal to 27 – the number of unit cubes, which can then be mapped on the largest gnomon. That is, all unit cubes can be put into one-to-one correspondence with 27 rectangles. The sum of products with the factor two is equal to eight – the number of 2x2x2 cubes, which can then be mapped on the next gnomon. That is, all 2x2x2 cubes can be put in one-to-one correspondence with another eight rectangles. Finally, the 3x3x3 cube can be mapped on the top left cell of the grid (Fig. 2.3). In that way, geometric objects of different dimensions can become connected by answering a second-order question using the combination of the first-order symbols (cubes and rectangles) and the second-order symbolism of the multiplication table. Similarly, one can consider grids and cubes of an arbitrary dimension n and recognize that, regardless of n, the number of rectangles is the same as the number of cubes, just as in the case n = 3 demonstrated above. Explorations with the first-order symbols integrated with the second-order symbolism can be enhanced by the use of technology. For example, formulas for the summation of the first, second, and third powers of integers (discussed in Chapter 7) can be developed first in the context of actions on concrete materials being abstracted to their algebraic representations which are then verified computationally and finally proved mathematically (e.g., by mathematical induction) – a typical experimental mathematics approach. In that way, technology has

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enormous cultural influence on the development of mathematical thinking. 2.4 Technology as a cultural support of problem posing The use of technology in mathematical problem posing can be characterized as a cultural support in designing new curriculum materials for a mathematics classroom. Through this creative process one uses tools of technology developed by advanced members of a technological culture for various practical and scholastic purposes. For example, a spreadsheet, originally developed for non-educational purposes, was retrofitted to be used in education, including mathematics education. In particular, in the specific context of mathematical problem posing, one puts to work the power of technology in order to formulate a gradeappropriate problem. Many questions, both of the first- and second-order, originate in the process of solving problems. Technology, when used appropriately, generates solution to a problem and, at the same time, provides data for a new problem to be posed. This implies that technology makes problem posing and problem solving being inherently linked to each other. The important task for a teacher candidate is to appreciate this relation between problem posing and problem solving and to learn how to recognize the emergence of a new problem. Such recognition should be developed both in a computational and noncomputational learning environment. Seeing technology as a cultural support of mathematical problem posing, one should note that the availability of powerful computational tools does not guarantee their appropriate application, unless one examines the effects of support in the context of using the tools [Cole and Griffin, 1980]. With this in mind, the ideas about didactic complexity of problem posing with technology, to a large extent, resulted from the author’s analysis of problem-posing activities by teacher candidates. As part of the activities, the latter were expected to critically reflect on the problems posed and discuss the role of computational environments (some of which were developed by the author) used to pose the problems. However, in order for technology to have a positive effect on problem posing, one should not only know how to use it but, more

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importantly, how to interpret the results that a technology tool generates. This interpretation requires understanding of what may be called the didactical coherence of a problem [Abramovich and Cho, 2015]. This notion can be applied to a non-digital problem posing as well. In what follows, the detailed description of these ideas will be presented. 2.5 Numerical coherence in problem posing Numerical coherence of a problem refers to its formal solvability either using given numeric data or within a given number system. Solvability may depend on different factors. Whereas with $10 one can pay for a $7 lunch, the two numbers may not be swapped; that is, with $7 one cannot pay for a $10 lunch. How can one formulate a question with the numbers 10 and 7 involved so that both subtractions become possible by extending the non-negative number system? For that, one has to create a context for using negative numbers. Weather is such a context: outside temperature can drop both by 7 and 10 degrees from, respectively, 10 and 7 degrees. Other contexts for negative numbers can be considered. Once again, the symbolism of negative numbers “remind[s] us of the phenomena which were the occasion for the formation of the new concepts” [Hilbert, 1902, p. 442]. That is, numerical coherence of a problem depends both on numbers and context involved. Furthermore, a number system associated with a problem depends on a grade level. For example, a problem of dividing two cakes among three children is not numerically coherent for the first graders who are familiar with whole numbers only. Yet, they can physically perform this division (aiming at fair sharing). Why does one need a number to describe the result that can be physically carried out? In other words, why does one need to move from the “first-order symbols … directly denoting objects or actions [to their formal representations through] the second-order symbolism” [Vygotsky, 1978, p. 115], which allows one to abstract from the concreteness of objects used in actions? One needs the second-order symbolism (e.g., fractions) when dealing with the first-order symbols (cakes and the action of dividing them into equal pieces) because the action is not the end of the story. Often, the results of several actions have to be quantitatively compared.

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For example, when does the division of cakes (of the same size) lead to a larger quantity for an individual: when two cakes are divided among three people or five cakes are divided among seven people? The division is possible at the level of the first-order symbols; yet quantitative comparison of the results is possible through the use of the second-order symbolism. Indeed, 5/7 is greater than 2/3 by 1/21 – a difference almost unrecognizable when pieces of cake are compared visually. At the same time, because two bicycles cannot be divided among three children, one cannot talk about 2/3 of a bicycle. (Although, to a certain contextual extent, one can talk about one-third of bicycle when three brothers share it claiming the ownership of the fractional part). Once again, this shows how numerical coherence of a problem depends on context. Furthermore, all the examples show the primordial role of physical action over its symbolic description. Sometimes, as in the case of geometry, numerical coherence of a problem depends on numeric characteristics of shapes to be constructed. For example, it is not possible to construct a triangle with the side lengths 2, 3, and 6 linear units. Here, however, the change of number system alone does not affect the problem’s solvability. Rather, to make this problem numerically coherent one has to choose three side lengths satisfying the classic triangle inequality – the sum of any two sides of a triangle is greater than the third side. By the same token, whereas on a geoboard it is not possible to construct a triangle with whole number sides touching four pegs and including one peg, such triangle with side lengths not being whole numbers can be constructed in more than one way (Fig. 2.4). That is, the change of number system makes the construction of triangles touching four pegs and including one peg on a geoboard possible. In particular, the possibility of this construction implies that the side lengths of such triangles satisfy the triangle inequality. This, in turn, raises an interesting question: is the number of non-congruent triangles touching four pegs and including one peg on a geoboard infinite?

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Fig. 2.4. Triangles touching four pegs and including one peg.

Numerical coherence of a problem can be obvious, like the problem of finding the sum of the first n natural numbers – whatever the answer is, it is a natural number because natural numbers are closed under the operation of addition. This is what the Common Core State Standards [2010] referred to as the use of structure by students as they learn to develop formal reasoning skills. By the same token, numerical coherence of a problem can be hidden and may not be easily verified without possessing a specialized knowledge. For example, a problem of constructing a triangle with integer side lengths on a geoboard touching eight pegs and including three pegs is solvable due to the Pythagorean theorem equating the square of the hypotenuse of a right triangle to the sum of squares of its legs. At the same time, the problem of representing the number 64 as a sum of consecutive natural numbers is not solvable because powers of two are the only natural numbers that do not have such a representation. Likewise, it is not possible to solve the trigonometric equation sinx = 2 for real x or the algebraic equation x3 + y3 = z3 in natural numbers, as well as to partition a prime number into the sum of three of more integers in arithmetic progression (see Chapter 4, section 4.5). So, numerical coherence of a problem is not a simple matter. 2.5.1 Using a spreadsheet to pose a numerically coherent problem Teacher candidates often seem to miss the point that problem posing “is a platform from which further development proceeds” [Davis, 1985, p. 23] and don’t go beyond posing a problem. Typically, earlier research on problem posing supported this point of view [Silver, 1994; Silver and

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Cai, 1996]. Some studies, however, suggested that the two activities – posing and solving a problem – are in the relation of dichotomy. One such study [Crespo, 2003] indicated that knowing how to solve a mathematical problem is not enough to be able to pose a problem. This view may be due to the fact that just as many young children traditionally see their role at school as an engagement in answering, not asking questions [Tizard, Hughes, Carmichael and Pinkerton, 1983], many teacher candidates believe that problem posing activity ends with posing a problem rather than being followed by its problem-solving phase. Nonetheless, it is possible for technology to be used to pose a problem in a way that its solution (or, better, answer) only requires correct interpretation of the results of computation. In what follows16, a spreadsheet will be used as a modeling tool enabling problem posing of numerically coherent problems. As an illustration, consider a problem posed by one of the elementary teacher candidates using a spreadsheet designed by the author to numerically model equations in three unknowns. Problem 2.1. Casey goes to the mall for her birthday! She goes to her favorite toy store where everything is on sale for $15, $10, or $5. If Casey has a total of $30, how many different ways can she spend it in the store? The spreadsheet of Fig. 2.5 (see Appendix for programming details) shows that there are seven ways to spend money in the store: the software counts the number of non-empty cells in the range D5:K9 and displays this number in cell A8. For example, the triple of numbers in the cells E4, C6, and E6 gives the following representation: 30 1 ∙ 15 1 ∙ 10 1 ∙ 5. One can say that Problem 2.1, the mathematical model of which is the equation 15x + 10y + 5z = 30, is numerically coherent and the triple (x, y, z) = (1, 1, 1) is one of the solutions. The teacher candidate who posed the problem explained the concept of numerical coherence as follows. 16 Problems included in this chapter were also discussed in the author’s paper [Abramovich, 2015a].

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Fig. 2.5. Spreadsheet solution to Problem 2.1.

My problem is definitely numerically coherent. $30 is a multiple of all the sale price amounts that I gave ($15, $10, and $5). If I had changed the amount to something different, say $27, then my problem would not be numerically coherent. It is very important that teachers review any problem they create and make sure that they are numerically coherent. Indeed, if one sets the content of cell A2 at 27, cell A8 would display zero reflecting the absence of numbers (the amounts of $5 toys, zeros included) in the range D5:K9. In other words, the equation 15x + 10y + 5z = 27 does not have integer solutions. The teacher candidate also attempted explaining how the spreadsheet of Fig. 2.5 works so that one can use the tool in solving the equation. In fact, elementary school students for whom this problem is designed are not expected to use a spreadsheet. Rather, the students are expected to either use trial and error or reason systematically. The latter type of reasoning is used by the teacher candidate in her analysis of how the modeling data of Fig. 2.5 corresponds to a possible paper-and-pencil solution. ... the first combination was if Casey bought two toys at $15 each: $15+$15; the spreadsheet displayed that. If I broke one of the $15 into $10 and $5, I would have a second combination: $15+$10+$5; the spreadsheet showed that Casey could buy one toy at each of these price levels. If I broke up the $10 into two five dollars, I would have my third

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combination: $15+$5+$5+$5. . . . [Finally,] I then split up the last $10 into five dollars and got my last combination: $5+$5+$5+$5+$5+$5; the spreadsheet showed that the [sic] Casey could buy six toys for five dollars each. In her analysis, the teacher candidate implicitly highlighted the idea of reduction (of a problem with three unknowns) to a simpler problem (with two unknowns), something that can be used when technology is not a part of the learning environment. One can see how one of the most powerful problem-solving techniques in mathematics as well as in mathematical didactics – recall the advice “solve a simpler problem” [Pólya, 1957, p. 114] – was incorporated into the very design of the spreadsheet of Fig. 2.5. Students can (and should be taught to) use this idea in their paper-and-pencil solution of Problem 2.1. 2.6 Contextual coherence in problem posing Contextual coherence of a problem comes into play when its solution has to be interpreted in terms of a context within which problem posing occurs. Besides the need to understand the context of a problem statement, it requires one’s appreciation of hidden assumptions grounded into one’s real-life experience and cultural background. As the teacher candidate, justifying contextual coherence of Problem 2.1 (she posed it for a grade three classroom), put it: My problem deals with birthdays and going to a toy store to buy toys. If the problem was for seventh or eighth grade, I could switch it to birthdays and going to a game shop. Also, I expect that my students all know what dollar bills we have in our country; however, I am aware that if I have a student from a different culture then they may not know how to make the different combinations that add up to $30. Note that young children can ask a variety of non-mathematical questions about Problem 2.1 such as: How old is Casey? Who gave her money? What kinds of toys were in the store? Nonetheless, answers to those questions by teachers can help the children better understand context in order not to ignore relevant or to use irrelevant information

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(cf. the travel problem presented in Chapter 1, section 1.1). Generally speaking, contextual coherence of a problem is a variable attribute. Just as without conceptual mastery of the base ten system (a cultural tool designed to support one’s arithmetic skills) a child cannot fully understand the meaning of a multi-digit number, without the mastery of other base systems a teacher candidate is not fully prepared to teach the base ten [Vygotsky, 1962]. Cobb’s [1995] writings on the learning of arithmetic emphasized the importance of the mastery of the numeration system as a cultural tool. In formulating problems in context, one has to be sensitive to the issue of culture, as errors that students make when solving problems may result from their own way of understanding context. One of the essential characteristics of mathematics is that its models are capable of describing diverse contexts within which multiple phenomena can be observed. In other words, different first-order symbols can be described through the same second-order symbolism (e.g., as shown in Fig. 2.2., the number of cubes within a cube is the same as the number of rectangles within a grid). This relationship underscores that the second-order symbolism is a cultural tool of abstracting from the first-order symbols, something that in the case of mathematics is practically universal across different cultures. One can see the same number written on the packs of different food items – cookies, candies, donuts; furthermore, this number may describe different characteristics of the content of the packs – quantities, calories, weight. Likewise, but at a higher level of abstraction, the equation 1/k + 1/m + 1/n = 1/2 describes several actions and objects: work done jointly by three workers in two days if each of them can complete the work in k, m, and n days, respectively; right triangular prisms with integer dimensions k, m, n and surface area numerically equal to volume; regular polygons having k, m, and n sides that enable edge-to-edge tessellation; triangles with the heights of lengths k, m, and n linear units into which a circle of radius 2 linear units can be inscribed. The last case is different from the other three cases because the equation (an algebraic model of a geometric situation) includes extraneous solutions representing segments which may not be used as the heights of a triangle. So, the model requires a refinement in terms of

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inequalities among the heights. These inequalities, necessary for the triangle inequality to be satisfied, provide an additional description of the first-order symbols enabling the construction of the triangle given its heights. At the same time, there are situations when the corresponding second-order symbolism hides a seemingly missing description of the relations among the first-order symbols within which a situation originates. For example, if Al is taller than Bob and Bob is taller than Chuck, the information that Al is taller than Chuck is correct, yet it is hidden. Using the second-order symbolism, the situation can be described through the inequalities a > b and b > c which hide the inequality a > c – the result of transitive inference from the first two inequalities. However, if Chuck is taller than Bob, then the relationship between the heights of Al and Chuck becomes undetermined. Such interplay between the conceptual and the procedural is an important factor to be considered in the context of problem posing in order to avoid the inclusion of extraneous information in a problem to solve. Although originally, mathematical model stems from a specific context, the mind constantly searches for other contexts that match the model which successfully describes this context. The ability to associate model with context and, vice versa, to match context with model, that is, to move freely from one type of symbolism to another, develops through problem posing. Context is very important for understanding mathematics. As was mentioned above, context can be the critical tool in dealing with abstraction and it can motivate a new perspective on numerical coherence. Indeed, while one cannot spend $10 having $7, one can comprehend the drop of temperature by 10 degrees from 7 degrees as the context of measuring outside temperature (in certain places of the world) already includes negative temperatures, the second-order symbolism for which does exist in the form of marks on a thermometer. Yet, in order to understand context, one needs to possess social competence, which may vary across different cultures and geographical locations. For example, while the above-mentioned drop of temperature is contextually coherent for a child in Alaska, it is not contextually coherent for a child in Singapore (regardless whether temperature is measured in Celsius or Fahrenheit). This requires the formulation of a

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problem to be consistent with cultural background and social competence of a mathematics classroom that nowadays often consists of ethnically amalgamated groups of pupils. This brings another type of coherence of a problem. 2.7 Pedagogical coherence in problem posing Pedagogical coherence of a problem includes such issues as attention to students’ on-task behavior, the absence of (or minimizing) extraneous data, the level of syntactic complexity [Silver and Cai, 1996], semantic/contextual clarity, grade appropriateness, and a method of solution expected. To explain the idea, following is an example of a pedagogically incoherent problem for a second-grade student; a problem, that manifests semantic ambiguity within a familiar context. A teacher, who, during her field experience, was assisting the student in his struggle with the problem, described this ambiguity in a weekly journal as follows. The problem posed the following scenario: “15 students went out for recess. 9 students did not play soccer. How many did play?” Immediately, the student concluded the answer to be 15. I wondered if he had not realized he needed to use subtraction. To respond, I read the question again. However, after the second reading, the student still saw no other possibility than 15. Then he made the comment, “It has to be 15, right? Because 15 went out for recess.” As he went on to repeat the comment twice more, I finally realized the student’s confusion: he saw the statement about not playing soccer as irrelevant. It was only after I clarified to the student that the question was asking how many played soccer that he understood; within seconds of this revelation, he deduced he needed to use subtraction and came to the correct solution: 6 students. One can see that semantic ambiguity of the problem did not allow the student to connect the numbers 15 and 9. Whereas the problem, talking about recess and soccer, was definitely contextually coherent for a second grader, it was not pedagogically coherent for him requiring additional clarification of the whole-part relationship between the

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numbers. It is only after the teacher candidate’s intervention that the student realized that the statement “9 did not play soccer” meant 9 is a part of 15 and, consequently, recognized in the situation a whole-part subtraction problem. This episode underscores the importance of the teachers’ shift in focus from their own teaching skills and abilities to what students are learning as a result of this teaching [Niess, 2005]. The notion of pedagogical coherence of a problem does enable such a shift. In the context of posing problems with technology, pedagogical coherence sometimes runs into conflict with an open-ended approach in the teaching of mathematics. The effectiveness of using this approach has been repeatedly emphasized in mathematics education research [Becker and Shimada, 1997; Kwon, Park and Park, 2006; Nohda, 1995; Pehkonen, 1995; Zaslavsky, 1995]. In particular, the use of problems with more than one correct answer is emphasized in open-ended pedagogical situations. However, “quite a few” numbers have the property of being greater than one. What do we want students to learn when offering them a problem with many answers? Do we want them to find all the answers? The ease of generating multiple answers to a single question in a computer environment turns a positive affordance of the learning environment into a negative affordance of educational computing affecting pedagogical coherence of a problem. Consider another problem posed by one of the teacher candidates with the help of technology. Problem 2.2. How many ways can one make 50 cents out of pennies, nickels, and dimes? Using a spreadsheet like the one shown in Fig. 2.5, yields the answer immediately – there are 36 ways to change 50 cents into dimes, nickels, and pennies. Whereas the modern pedagogical dictum “single question – multiple answers” helps young children to appreciate the concept of multiplicity of answers in a mathematical problem, posing a question with 36 answers does not make much sense. Teacher candidates are often confused who uses technology, when, and why it is used. In the real classroom, it is only teachers who are using technology for posing problems. Yet, their students will be working on a task like Problem 2.2

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using pencil-and-paper alone and, thereby, a problem with so many answers, defies itself “clearness and ease of comprehension” [Hilbert, 1902, p. 438]. The teacher candidate who posed Problem 2.2, correctly described this issue through the following statement: I would not expect young children to find all of the solutions to my problem. It would take a very long time to solve the problem with the numbers provided to them. She, however, goes on to make a conflicting remark regarding the problem’s pedagogical coherence: I believe my problem is pedagogically coherent because I feel the students will be able to find most of my 36 solutions. It should be fun for students to click around and experiment with different solutions. It appears that the teacher candidate was too excited about her problem when used the word most, meaning, perhaps, some (solutions). Notwithstanding, the idea of experiment in the context of using a spreadsheet is a good one because it points to the fact that one can effectively revisit pedagogically incoherent problem in order to formulate a problem with the single answer. Towards this end, one can be asked to use a problem with a large number of computer-generated solutions (answers) in order to formulate a problem with the single answer. Thus, selecting a particular solution may be a way of turning a pedagogically incoherent problem into a coherent one. After a classroom discussion of what such experimentation with a pedagogically incoherent problem of that type may entail, another teacher candidate has formulated a problem with the single solution by correctly interpreting data generated by a spreadsheet. Problem 2.3. Carol spent exactly 50 cents buying some 10-cent stamps and as many 5-cent stamps as 1-cent stamps. How many stamps of each denomination did she buy? In posing this problem, she noted that one of the ways to partition 50 into the summands 10, 5, and 1 is through the equality 50 2 ∙ 10 5 ∙ 5 5 ∙ 1. Reflecting on the posed problem she wrote: In my classroom, I will use this question to help the students to begin to form and solve a system of algebraic equations. [Indeed, such a

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system has the form 10x + 5y + z = 50, z = y, leading to the equation 5x + 3y = 25 with the single solution x = 2, y = 5]. These types of questions also lead to a discussion of LCM [least common multiple] and GCF [greatest common factor], changing the total amounts and the amounts of the price of the stamps. The teacher candidate didn’t explain how the concepts of LCM and GCF could be connected to Problem 2.3. The author can only guess what her thinking had been: if the largest denomination were 9-cent, then the problem is not numerically coherent. Indeed, in the resulting equation, 9x + 6y = 50, whereas GCF(9, 6) = 3 we have GCF(3, 50) = 1. (Alternatively, LCM(50, 3) = 50·3, indicating that 50 is not divisible by 3). This slight modification is different from Problem 2.3 where in the resulting equation, 5x + 3y = 25, we have GCF(5, 3) = 1 and GCF(1, 25) = 1. In her unexplained comment, the teacher naturally touched upon Dewey’s [1938] pedagogical construct of collateral learning. To conclude this section, note that pedagogical coherence of a problem depends on the expected method of solution. Often, as students learn using more and more sophisticated mathematical tools, a numerically and pedagogically incoherent problem for a lower-grade level (like dividing two cakes among three children) becomes numerically and pedagogically coherent for an upper grade level. The opposite relationship can be observed as well: a pedagogically coherent problem for a lower grade level may become pedagogically incoherent for an upper grade level. For example, whereas for a six-year-old pupil (who uses concrete materials – the first-order symbols – as means of problem solving) the tasks of arranging 24 students and 25 students into four groups to do a team work are at the same level of complexity, for a ten-year-old pupil (expected to deal with the operation of division as the second-order symbolism) the latter case is conceptually more difficult for it requires the interpretation of the meaning of remainder. Likewise, for a middle school student, the problem of finding on a geoboard the rectangle of the given perimeter with the smallest area is didactically coherent – such rectangle has the smaller side length equal one linear unit. However, for a high school student such a problem when posed in the Euclidean plane is both numerically and pedagogically incoherent. These examples demonstrate the complexity of interplay that exists

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among contextual, pedagogical, and numerical coherences of a problem. Furthermore, as will be shown later in the book, in the digital age new methods of resolving mathematical queries emerge, thus adding another dimension to the notion of pedagogical coherence. 2.8 Didactical coherence in problem posing The above three coherences can be presented in the form of a Venn diagram (Fig. 2.6). It shows that ideally, a problem posed to students has to belong to the intersection of the three coherences where the problem becomes didactically coherent. For example, the problem about recess and soccer, considered through the lenses of the second grader, belongs to region A (familiar context and numeric solvability); yet it is only the teacher’s intervention that made the problem pedagogically coherent and placed this problem verbal modification into the didactical coherence region. Attending to the notion of didactical coherence of a problem allows for a greater effect of computational support on problem posing. The Venn diagram, itself being a cultural tool, didactically supports technology-enabled problem posing. The diagram can be seen as a thinking device that scaffolds teacher candidates’ ability using “comprehension-fostering and self-regulatory cognitive and metacognitive strategy … [through] question generation” [Yu, 2009, p. 1135]. By learning to use the Venn diagram as a tool that informs problem posing, one develops higher order thinking and reasoning skills and gains valuable research-like experience in preparing curriculum materials. Consider region C in the Venn diagram of Fig. 2.6. It may include numerically incoherent, yet pedagogically and contextually coherent problems. For example, within the non-negative number system, one cannot get a $10 toy by paying $7. The change of context from a toy store to a four-season climate brings about negative numbers enabling one to find temperature after it dropped by 10 degrees from 7 degrees thus placing so modified problem within the didactical coherence region. Likewise, within the whole number system one cannot compare prices of a single piece of cake when five cakes are divided equally among seven people versus two cakes (same size) divided equally among three people. While dividing cakes provides appropriate context and pedagogy, unless

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the number system is extended to fractions, the prices do not have numerical meaning. Finally, consider region B of the Venn diagram of Fig. 2.6. It may include two types of problems. Within this region, a problem may become numerically coherent for a higher-grade level (e.g., solving the quadratic equation x2 + x + 1 = 0 in the set of complex numbers) or a problem may become pedagogically coherent for a lower grade level under a weaker condition on the variables involved (e.g., finding triangular numbers comprised of the same digits only when the number of digits is not greater than two – see Chapter 1, section 1.5.1). Yet both problems are missing a real-life context. This, of course, does not mean that all numerically and pedagogically coherent problems in school mathematics have context. Nonetheless, at any grade level, the art of posing mathematical problems, which, historically, “spring from experience and are suggested by the world of external phenomena” [Hilbert, 1902, p. 440], includes skills of embedding formal mathematics into a real-life content.

Fig. 2.6. Regions A – F and Didactical Coherence have empty intersection.

2.9 Summary

By reflecting on a number of scholarly writings by notable educators of the 20th century, this chapter revealed that problem posing, in general, and mathematical problem posing in particular, has been considered as

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educational philosophy. Such perspective on problem posing was first put forward by Maria Montessori in Italy and then echoed in the writings by Tsunesaburo Makiguchi in Japan and Paulo Freire in Brazil. Another theoretical aspect of mathematical problem posing, especially in the presence of digital tools, deals with the need for a problem to be didactically coherent in a number of respects, despite or perhaps because of the ease of generating problem data by a computer. Three major components of didactical coherence were introduced and discussed through the analysis of problems posed by elementary teacher candidates using a spreadsheet. A point was made that problem posing and problem solving are two sides of the same coin that teachers use to teach and students use to learn mathematics. The next chapter will introduce the notion of technology-immune/technology-enabled problems. Under the umbrella of this notion, it will be shown how the use of technology motivates new ways of developing problem-solving curricular materials for the modern-day mathematical classroom

Chapter 3

Posing Technology-Immune/TechnologyEnabled (TITE) Problems 3.1 From teaching machine movement to symbolic computations In order to demonstrate how the advent of different technological tools into the classroom has changed the ways teacher candidates can learn mathematical problem posing, this chapter begins with a brief historical review. In the United States, the use of computers in education has its roots in the teaching machine movement stemming from the work of Pressey [1926] who believed “mechanical aids are possible which ... would leave the teacher more free for her most important work, for developing in her pupils fine enthusiasms, clear thinking and high ideas” (p. 376). A Pressey’s machine, with its main focus on allowing students to move at their own pace by receiving immediate feedback to a multiple-choice test (which could include a mathematical task), has provided them with the opportunity to become active participants in the learning process [Skinner, 1968]. Yet, only in the second part of the 20th century these pioneering ideas about the power of automated instruction had evolved into the major research and development agenda concerning programmed learning [Lumsdaine, 1960]. In particular, the use of programming languages such as BASIC (e.g., [Roberts and Moore, 1984]) or PASCAL (e.g., [Carmony, McGlinn, Becker and Millman, 1984]) in the context of teacher education made it possible to teach mathematics with the flavor of exploration and discovery. This shift in mathematics pedagogy from procedural to conceptual, in addition to mathematics teaching, required, as Demb [1973] put it, ‘teaching about’ rather than ‘teaching with’ the computer. Later, technological advances reduced the need for computer programming, thus allowing for ‘teaching [mathematics] with’ the computer [Shaw et al., 1997; Garofalo, Drier, Harper, Timmerman and Shockey, 2000]. Among such advanced applications that are relatively user-friendly, in order to be utilized in teacher education and, consequently, in pre-college contexts, are various

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dynamic geometry programs, computer graphing software, computer algebra systems, and electronic spreadsheets. The minimal amount of programming still required for teaching mathematics with a computer is mostly used as a way of expressing and articulating mathematical ideas through a computational medium [Noss and Hoyles, 1996]. In many cases, the medium serves as an agent of problem-solving activities that underlie numeric and/or graphic modeling of a mathematical situation under study. For example, in order to graph the segment [-1, 1] of thickness 2in the context of the Graphing Calculator, one has to define the segment through the system of inequalities | x | ≤ 1, | y | ≤ (see Chapter 5 for more detail). That is, mathematics teaching can both be supported by technology and integrated with its use. More recently, new developments in mathematical modeling software tools enabled symbolic computations, a traditional component of doing mathematics at different levels of sophistication, to be effectively carried out by a computer. As a result, such technological advances not just opened new research opportunities for professional mathematicians under the umbrella of experimental mathematics [Epstein, Levy and de la Llave, 1992; Borwein and Bailey, 2004; Arnold. 2015], but, quite unexpectedly, created new challenges for mathematics educators. One such challenge is that a computer can now answer even a loosely formulated question (i.e., a question being linguistically coherent but mathematically colloquial). Due to informal structure of such a question, a computer might offer several options for a student to select the correct answer. Sometimes, one’s ability to navigate through these options requires knowledge of mathematics beyond the grade level involved. Furthermore, mathematical notation may vary from tool to tool thus making a student dependent on a particular coding practice. For example, the notation INT is understood by Excel as the greatest integer function, is ignored by the Graphing Calculator [Avitzur, 2011], and is treated by Wolfram Alpha as integral (antiderivative). So, typing “INT(3x-1)=4” into the input box of Wolfram Alpha instructs the program to integrate 3x – 1 from 0 to x and equate the result to the number 4. This yields the equation 4 or 3 2 8 0 with the roots

and x = 2. The two values of x do not satisfy the

original equation as INT 3 ∙

1

5 and INT 3 ∙ 2

1

5.

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At the same time, replacing INT by its alternative notation FLOOR and typing “FLOOR(3x-1)=4” yields the equation 3 5, thereby, still leaving it to the user to find x (even if the notation 3 is understood correctly. Moreover, the input “FLOOR (3x-1)=4” with a space separating the word “FLOOR” from the expression “(3x – 1)” is not understood by the program. However, if the user is able to appreciate the meaning of the lack of expected response produced by the software and enter an augmented request “solve the equation FLOOR(3x-1)=2”, thus making “a switch from one system of semiotic text awareness to another at some internal structure level” [Lotman, 1988, p. 43], Wolfram Alpha yields the correct solution in the form of the inequality 5/3 ≤ x < 2. In that way, in the context of Wolfram Alpha, the values x = -4/3 and x = 2 when selected as solutions to the equation INT(3x – 1) = 4 are irrelevant, leaving the answer in the form 3 5 represents an incomplete solution which does not demonstrate understanding of what is expected from an answer, and having space between the notation FLOOR and the expression (3x – 1) leaves the program confused. A possible variance in human-computer interaction emphasizes the need for one’s grasp of conceptual meaning of the procedures involved in computing and their expected outcomes. In turn, this requires a combination of conceptual understanding and procedural skills as pillars of mathematical problem solving. One of the creative approaches to bridging the conceptual and the procedural, in the milieu of a problemsolving uncertainty, is the ability to “check the result” [Pólya, 1957, p. 59]. In the author’s own pre-college learning experience, verification of the correctness of a solution/answer to a problem was a mandatory conclusion for almost any mathematical task. This verification can be done either through the plug-in strategy (for a numeric answer) or by recourse to a special case (whatever the nature of uncertainty). In the case of the greatest integer function which returns the largest integer not exceeding a given number, one can enter into the input box of Wolfram Alpha the command “INT (1/2)” and expect the program to return zero rather than x/2 (the result of integration). Otherwise, the basic conceptual understanding of the notation used would suggest it has a different meaning for the program which, thereby, delivered an incorrect result. Likewise, even the direct verification of an answer through the plug-in

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strategy, while being purely procedural, may bring about a counterexample to one’s conceptual assumptions. It appears that the diversity of thinking encouraged by the current standards for teaching mathematics might be ill advised by a computer. By the same token, many traditionally formulated problems can be solved unambiguously by software with a minimal contribution by a student. This raises the issue of the assessment of work carried out on a computer. In the presence of what Guin and Trouche [2002] called “an automatic transport phenomenon ... [when the outcome of problem solving process depends on whether] one can feed all the problem’s data into the machine” (p. 205, italics in the original) new instrumentation processes have to be used in order to boost students’ contribution to mathematical problem solving. In other words, new types of problems have to be created. This motivates new perspectives on the technologyenabled problem posing17. 3.2 Technological advances call for the revision of mathematics curriculum In the 1970–1980s, outside the context of using LOGO at the elementary level [Feurzeig and Lucas, 1972; Noss, 1987], the didactic emphasis on technology integration in mathematics education was mostly on drill and practice in arithmetic and basic shape construction in geometry. In the general context of education, Maddux [1984] referred to such perspectives on teaching with computers as Type I application of technology advocating for Type II applications as “new and better ways of teaching” (p. 38, italics in the original). The most common Type I applications of technology to mathematics education are those that support two pedagogical foci – drill and practice and entertainment. On the contrary, solving multistep problems, using mathematical concepts as tools in computing applications, or exploring curricular topics that otherwise are not attainable are examples of Type II application of technology in mathematics education. The notion of the two types of technology application underscores the difference between instructivist and constructivist learning environments that measure learning, respectively, by tests and by one’s 17 Problems included in this chapter were discussed in the author’s papers [Abramovich, 2015a, 2018].

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ability to systematize, analyze, guess, and invent as a way of wrestling with big ideas [Brooks and Brooks, 1999]. Unlike instructivist learning environments that are concerned with a particular way of knowing and knowledge construction, the role of constructivist learning environments is to support multiple ways of knowing in order to help learners develop the so-called epistemic fluency [Morrison and Collins, 1996] – the ability to comprehend and use different ways of knowing and knowledge construction. Just as the boring and clumsy uses of mathematics could (and should) motivate the development of its more procedurally effective concepts, one’s mundane experience with Type I application of technology can motivate the development of creative ideas to support educational applications of Type II. For example, repeated addition (subtraction) motivated multiplication (division), which, in turn, motivated the use of logarithms. Nowadays, if one is asked to find the remainder when dividing 1266 by 32, an electronic spreadsheet can be used as a commonly available computing device. The operation of division yields the number 39.5625, with remainder being hidden in the decimal representation of the result. A kind of a clumsy (yet conceptually correct) way to get the remainder would be to subtract 39 from the quotient to get 0.5625 and multiply the latter by 32 to get 18. This computation can be carried out through a simple spreadsheet formula =(1266/32-INT(1266/32))*32, where INT is the greatest integer function (correctly understood by a spreadsheet). Repeated subtraction of 32 from 1266 is boring and ineffective for it requires 39 steps to reach the remainder 18, although it can also be computerized by entering 1266 in cell A1, defining in cell B1 the formula =IF(A1=” “,” “,IF(A1-32>0,A1-32,” “)) and replicating it across row 1 until the operation of subtraction ceases providing positive difference (Fig. 3.1).

Fig. 3.1. Obtaining remainder through repeated subtraction using a spreadsheet.

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A less cumbersome approach to computing the remainder is to use the following algorithm: 1266÷32 = 39 + 0.5625, 1266 = 32·39 + 32·0.5625, 1266 = 32·39 + 18. The last equality is the fundamental relation connecting dividend (1266), divisor (32), quotient (39), and remainder (18). Finally, the use of the MOD (alternatively the remainder) function (available in the tool kit of computing devices of a spreadsheet) through the formula =MOD(1266, 32) yields the remainder 18 immediately. That is, knowing the existence of the remainder function within a spreadsheet provides not only an example of Type II application of technology, but also induces learning about modular arithmetic, something that is a part of a number theory course. In the words of one of the author’s students enrolled in a teacher education course: “Concepts explored in our class were also explored in my number theory course. These classes actually covered some of the same concepts and each helped me with the other. I feel that explorations made possible by a spreadsheet did enhance my understanding of some of the mathematical ideas which seem a little bit clear now. This [teacher education] class helped me with understanding of the MOD function.” Nonetheless, one can say that the ease of using the MOD function in computing remainders does cross the boundary between two types of technology applications and it leaves the question open to which type such use of technology belongs. Indeed, as the sophistication of software products progresses, mathematical problem solving with the use of technology that until recently was considered its Type II educational application may become less and less cognitively complex. The use of the MOD function in finding a remainder is such an example. This reduction in complexity blurs the distinction between the two types. Put another way, it enables the corresponding epistemic game – “the set of rules and strategies that guide inquiry” [Collins and Ferguson, 1993, p. 25] within a particular representational structure – to be reduced essentially to a simple push of a button. In that way, the type of technology application may depend on what kind of technology is used to support the game. Furthermore, the sophistication of software products calls for a new mathematics curriculum which allows for both cognitive engagement and technology use as essential elements of mathematical problem solving. That is, the

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need for cognitively robust problem solving motivates the development of new ideas for problem posing. As another illustration of ambiguity in deciding the type of technology application, consider a classic problem of finding the sum of the first 100 natural numbers. According to a legend (e.g., [Dunham, 1991]), Gauss18 found the answer, 5050, almost immediately after the problem was offered to him in the primary school. The teacher of Gauss saw this problem through the procedural lenses, thereby expecting the class to be quite busy when adding 100 numbers one by one. Yet, as the legend goes, young Gauss demonstrated epistemic fluency in solving the problem by recognizing in the sum to be found a multiple of the arithmetic mean of the first and the last addends. That is, adding the sum sought twice and pairing numbers equidistant from the beginning and the end of the sum, resulted in the number 5050 = 101·100/2. Besides its historical appeal, this legend can serve as a springboard into a variety of technology-related explorations, some of which are discussed elsewhere [Abramovich, 2012] under the conceptual umbrella of collateral learning [Dewey, 1938]. For example, in the digital era, one can use a spreadsheet to generate partial sums of the first n natural numbers for different values of n. Such use of a spreadsheet, in order to be efficient, requires a certain level of mathematical sophistication that includes one’s understanding that the natural number sequence develops recursively and, therefore, each term of the sequence can be defined as the previous term increased by one. Also, in this computational environment, one can observe and then verify numerically, that the sum of the first n consecutive natural numbers is equal to n(n + 1)/2. Using a spreadsheet in that way is kind of cognitively demanding and, therefore, it may be considered as a Type II application of technology in mathematics education. However, those familiar with other computational tools might disagree. Indeed, typing “1+2+3+...+100” in the input box of Wolfram Alpha or even posing the query “What is the sum of the first 100 natural numbers?” which, perhaps, word for word repeats the question allegedly 18

Carl Friedrich Gauss (1777–1855) – a German mathematician, commonly regarded as the greatest mathematician of all time.

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posed by the teacher of young Gauss, yields the number 5050. That is, the same problem, depending on the kind of technology used, requires different levels of mathematical and technological sophistication. Unlike the case of using a spreadsheet, in the context of Wolfram Alpha one only needs to more or less carefully formulate a question and the program would deliver an answer at the push of a button. This apparent easiness of action, to a certain extent, may resemble Type I application of technology. That is, a switch from one computer program to another has turned Type II into Type I. 3.3 Definition of a TITE problem and a simple example The concept of technology-immune/technology-enabled (TITE) problem was introduced in [Abramovich, 2014] as an extension of the Type II vs. Type I technology applications framework. This dichotomy between the two types made it possible to justify the need for moving away from drill and practice (Type I) towards using technology as a conceptual tool (Type II). In mathematics education, the suggested extension is necessary due to what may be considered as a negative affordance of technology even when its Type II applications are the main foci of instruction. The ever-increasing sophistication of mathematical software tools, enabling one either by the design of a tool or by his/her technological skills to solve various multistep problems at the push of a button, can significantly reduce the traditional complexity of problem solving. This unintentional consequence of otherwise positive affordances of technology blurs the distinction between the two types. In order to sustain educational gains from the Type II versus Type I framework in the context of mathematics, a teacher candidate has to be proficient in designing tasks that are still cognitively challenging despite (or perhaps because of) the available power of symbolic computations and graphic constructions. By learning to pose and solve TITE problems, a teacher candidate develops the appreciation of the so-called “instrumental genesis” [Rabardel, 1995; Guin and Trouche 1999; Lonchamp, 2012] through balancing positive and negative affordances of high-level computer programs to enable Type II applications. Unlike the above-mentioned classic problem of finding the sum of the first 100 natural numbers, TITE

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problems cannot be automatically solved by software; yet, the role of technology in dealing with those problems by learners remains critical. As an aside, using TITE problems can open a new window on utilizing technology for the assessment of problem-solving skills rather than banning technology use in exams. In that way, the context of the TITE problem solving may be considered as Type II application of technology of the second order. Recall that in Chapter 1 (section 1.3) a spreadsheet was used to find the number 216 – the cardinality of the sample space of the experiment of rolling three dice and recording the sum of spots on their faces. This computation may be considered a TE part of the problem of finding this number which has to be followed by a TI part – providing a formal justification of this number using the rule of product. Nečesal and Pospíšil [2012] made a similar point in the context of teaching engineering calculus when applying Wolfram Alpha to nonalgorithmically solvable problems arguing that whereas a student is cognitively responsible for the whole problem-solving process, some part of this process can be outsourced to technology. In mathematics teacher education, an ability to pose (and solve) a TITE problem in the context of a digital tool (including Wolfram Alpha) can also be put in context of the technological pedagogical content knowledge (TPCK) framework [Mishra and Koehler, 2006; Niess, 2005] for it constitutes an important skill allowing teacher candidates to “advance from novice to expert thinking about designing instruction with technology” [Angeli and Valanides, 2009, p. 162]. Many problems in number theory and algebra can be posed in the spirit of the TITE concept. For example, whereas Wolfram Alpha cannot explicitly respond to the (inverse) query “Can the sum of the first n natural numbers be equal to 5050?” or to the request “Construct a biquadratic equation with the roots -3, -2, 2, and 3”, the program, notwithstanding, can handle, respectively, the tasks “solve n(n + 1)/2 = 5050” or “expand (x + 3)(x + 2)(x – 2)(x – 3)”. The formulation of the former task requires either knowing or learning through interaction with the software the meaning of the left-hand side of the equation to be solved – the sum of the first n natural numbers is equal to n(n + 1)/2. Furthermore, one has to keep in mind the meaning of the symbol n in

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order to reject its negative value as an extraneous solution. Alternatively, one can type “solve n(n + 1)/2 = 5050 for n > 0” to avoid the extraneous solution, thus demonstrating how conceptual understanding of a mathematical situation turns into a procedural skill of controlling symbolic computations. That is, the process of solving an equation by a computational medium becomes informed by the conceptual meaning of the symbol involved [Tall et al., 2001]. Similarly, the formulation of the latter task (i.e., constructing the equation 0) requires knowing the Factor Theorem. Alternatively, the query “81 + 9a + b = 0, 16 + 4a + b = 0” entered in the input box of Wolfram Alpha yields the values of coefficients of the biquadratic equation sought. Therefore, “being able to think about the symbolism as an entity, allows one to manipulate the entity itself, to think about mathematics in a compressed and manipulable way, moving easily between process and concept” [ibid, p. 88]. 3.4 Revisiting classic problems in the digital era under the umbrella of the TITE concept This section will provide examples of rather involved TITE problems which stem from the reformulation of known problems found in a classic book written in the pre-digital era. To begin, consider the task of guessing “the rule according to which the successive terms of the following sequence are chosen: 11, 31, 41, 61, 71, 101, 131, ...” [Pólya, 1954, p. 8]. One can note that all the numbers in this sequence have the same last digit and diminishing them by the number 1 yields a multiple of ten. This observation prompts the next question: which numbers with the same property are not included in this sequence? The sequence 21, 51, 81, 91, 111, 121, … then results. How are the two sequences different? A possible answer to this question is that whereas the terms of the latter sequence have more than two different divisors, the terms of the former sequence have exactly two different divisors and, by definition, represent prime numbers which are one greater than a multiple of ten. The task may be continued by asking for the next few primes of that kind. Which of the numbers 141, 151, 161, … are prime numbers?

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Nowadays, the last question as well as the original task can be outsourced to Wolfram Alpha, which, in turn, refers to the Online Encyclopedia of Integer Sequences (OEIS®, http://oeis.org) where one can find out right away that the first interpretation (among many others) is that the sequence 11, 31, 41, 61, 71, 101, 131, … represents prime numbers of the form 10n + 1. Curious students may attempt to study other interpretations of the sequence and this raises an issue of how one can use properly what is available when technology affords an easy access to large quantity of information [Conole and Dyke, 2004]. From the TITE perspective, the Pólya’s task may prompt asking the question: What is the smallest number of divisions one needs in order to decide whether each of the numbers 131, 141, 151, 161, … is a prime or not? One can use technology in making those divisions; yet technology, in the absence of mathematical reasoning, does not give an answer about the smallest number of divisions (except 141 which is an odd number divisible by 3 as its sum of digits is a multiple of 3; that is, requiring a single division). One can also be introduced to the on-line sieve of Eratosthenes [Meyer, 2016] and even do all divisions using this tool. In doing so, one has to realize that after 131 and 151 survived divisibility by 11 (as well as by 2, 3, 5, and 7), no division is necessary and both 131 and 151 are prime numbers. Moreover, one can use Wolfram Alpha to generate more prime numbers through the command “primes from 11 to 531 of the form 10n + 1” (Fig. 3.2).

Fig. 3.2. Generating primes of the form 10n + 1 using Wolfram Alpha.

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Fig. 3.3. Using Wolfram Alpha in solving a problem from a classic book.

A more difficult task from the same book [Pólya, 1954, p. 31] is to . This task can also be find the value of the infinite product ∏ 1 outsourced to technology. Entering the command “find the product (1from i=2 to infinity” into the input box of Wolfram Alpha yields 1/2 and the value of the partial product ∏

1

in the fractional form

(Fig. 3.3). It is not difficult to write down several numeric values of this algebraic expression:

, , ,

. One can be asked: Why is every

second fraction in this sequence represented in the simplest form? In the case of other fractions, is the number 2 always the greatest common factor between the numerator and denominator? Why or why not? What is the probability that the fraction of the form is irreducible when the natural number n is chosen at random? Technology cannot answer such (conceptual) questions. That is, seeking the values of the infinite and partial products is a request for information, something that can be outsourced to Wolfram Alpha. But the questions requesting explanation of numeric properties of an algebraic expression that represents the sequence of the partial products turns Pólya’s rather complex task into a relatively simple TITE problem. Through solving this problem, one can, in the spirit of Tall et al. [2001], recognize the duality of the fraction (or any other algebraic expression for that matter) – first using it as a process in generating numeric values for obtaining information and then

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seeing it as a concept when requesting explanation of specific properties of the sequence that have been revealed through exploring patterns formed by its numeric values. Another TITE problem might be to connect the infinite product and converging to 1/2 as well. For example, one can the infinite sum ∑ be asked to carry out the following explorations: ; using Wolfram Alpha find ∏ 1 using Wolfram Alpha find ∑ prove that

∏

1

; and

∑

are, respectively,

monotonically decreasing and increasing sequences; using a spreadsheet find N such that for all n > N the inequality | holds true for 0.01 and 0.001. |

Fig. 3.4. Modeling partial products and sums within a spreadsheet.

Note that through the suggested use of a spreadsheet one can converges to 1/2 faster than (Fig. 3.4, see Appendix conclude that for programming details). Why is it so? Even if one uses Wolfram Alpha

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to establish the relation ∑

, one still has some explaining to

do regarding the difference between the infinitesimal sequences

and .

In that way, a TITE problem consists of both first- and second-order questions: an informational type question that can be outsourced to technology and an explanatory type question that requires one to connect procedural and conceptual knowledge. A similar task from Pólya’s [1954, p. 31] book is to find the value of the infinite product ∏ 1 . Slightly modifying the last expression by substituting i + 1 for i and entering “find the product 1

from

i = 2 to infinity” into the input box of Wolfram Alpha yields 0.166667 as an approximation and the value of a partial product (e.g., when infinity is replaced by n) in the fractional form (Fig. 3.5). Note that the suggested modification of the form of the product is an example of how mathematical knowledge (referred to in Chapter 4 as advanced conceptual understanding) can be used to correct some negative affordances of technology when information it generates is not well suited for the goal of instruction – to formulate a didactically coherent TITE problem. Entering the product without modification would yield the fraction



not defined for n = 1 thus complicating some

issues which can be avoided by using technology in a creative way. That is, advanced conceptual understanding of mathematical problem posing enables for a smooth transition to the TI phase of the TITE problem solving. , an expression One can be asked to investigate the fraction defined for n = 1, 2, 3, ... . Through this investigation, the expression can be connected to triangular numbers and thus can be replaced by

, as

. The first TITE

question to be formulated here is: What is the greatest common divisor (GCD) of two triangular numbers separated by another triangular number? A simple spreadsheet investigation (Fig. 3.6, see Appendix for programming details) gives an answer to this question requesting information: the sequence GCD , forms the cycle (1, 1, 3) where

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the number 3 appears each time when n is a multiple of three. Now, one can be asked to explain this phenomenon conceptually. To this end, one is equal to 2n + 3, so that when n can show that the difference is a multiple of three, the difference is not only divisible by three but GCD , 3; otherwise, GCD , 1. One can see that the inquiry into the behavior of the sequence of the greatest common divisors is a TITE problem: the property of the sequence forming the cycle (1, 1, 3) is technology motivated and its explanation requires formal demonstration (a TI part) which, if necessary, may be enhanced by technology (a TE part).

Fig. 3.5. Another problem about infinite product.

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Fig. 3.6. The sequence

,

forms a cycle of period three.

Similarly, using a spreadsheet, one can try to explore the behavior of the sequence GCD , only to discover the absence of an interesting pattern19. Likewise, no simple pattern stems from the greatest common divisors of other pairs of triangular numbers. Was the connection of the infinite product to triangular numbers accidental? To answer this question, one can be asked to explore another problem from Pólya’s [1954, p. 32] book and to find the infinite product ∏ using Wolfram Alpha. In response, the program 1 generates the value of a partial product in the form

. This

time, using a spreadsheet, one can see that the sequence GCD 4 1, 2 3 2 5 for n = 3, 4, 5, .... forms the cycle (1, 1, 3) already observed in the previous example. This computational finding suggests at least three things. First, the cyclic behavior of the sequence of the greatest common divisors of the elements of the partial products is, perhaps, due to the products themselves and the observed connection to triangular numbers appears to be accidental. Second, formulating a coherent TITE problem requires understanding that such a problem should allow for a reasonable extension. Third, mathematical creativity 19 In Chapter 7, section 7.4, search for a pattern formed by the GCDs will be extended to the triangular number sieve of order one (introduced in Chapter 1, section 1.5.2).

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in posing TITE problems can be a motivating factor for the formal study of a mathematics content that was used as background for such problems. In particular, one can become motivated to study infinite products in a way it was designed in the classic book by Pólya [1954] written in the pre-digital era. Alternatively, triangular numbers can be investigated along the lines suggested in Chapter 1, section 1.5.2. Examples of such investigations provided in Chapter 7 would demonstrate that cyclic behavior of the GCDs observed above is not necessarily accidental. 3.5 Conceptual bond and arithmetical word problems 3.5.1 Looking at the past to develop new teaching ideas In the presence of technology, mathematical ability to pose a problem can be developed in the context of arithmetical word problems that bear a TITE flavor. Although arithmetic is commonly described as a window to algebra, arithmetical problem solving and posing become part of advanced school mathematics when conceptual understanding is expected to supplant algebraic routines provided by the use of variables. As was already mentioned in Chapter 2, Freire’s [2003] critical education theory emphasized that “problem-posing education … corresponds to the historical nature of humankind … for whom looking at the past must only be a means of understanding more clearly what and who they are so that they can more wisely build the future” (p. 84). By looking at the past, one can recall elementary school mathematics curriculum when students were expected not only to “check the result” [Pólya, 1957, p. 59], as mentioned in section 3.1, but to solve word problems without using algebra. Instead, their solution process was rooted in asking conceptual questions to be answered in a purely numeric form. One such problem can be found in the mathematics curriculum of the 19th century as described by Tchekoff20 [1970, p. 70] in a story Tutor: “If a merchant buys 138 yards of cloth, some of which is black and some blue, for 540 roubles [sic], how many yards of each did he buy if the blue cloth cost 5 roubles [sic] a yard and the black cloth 3?” To solve this 20 Anton Pavlovich Chekhov (alternatively Tchekoff) – a Russian short story writer of the 19th century (1860-1904).

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problem without using algebra (something that the tutor could not do), one can begin with “guessing” any additive partition of 138 in two positive integers, e.g., 138 = 100 + 38, and then proceed to calculating the payment that would have been made under this guess. In doing so, the sum of products of the prices and the meters, 3 ∙ 100 5 ∙ 38 490, has to be subtracted from the actual payment, 540, to get 50. The next consideration is that the difference between the actual and guessed payments has to be a multiple of the difference in prices for a yard of each type of cloth, 5 – 3 = 2. Therefore, 50 2 25 is an error made in the guessed partition of 138. This makes it possible to offset the original guess through subtraction and addition: 100 – 25 = 75, 38 + 25 = 63. That is, the merchant bought 75 yards of the black cloth and 63 yards of the blue cloth. 3.5.2 Posing similar problems How can one pose problems of that type? How can one create similar didactic materials in order to support the notion that “often a child left to himself will go back to the same puzzle he solved yesterday, simply for the pleasure of getting it right” [Mayer, 1965, p. xxxii]? Even if a strategy of solving a problem is known, a cursory change of data does not lead to a solvable (in integers) problem. For example, replacing 138 by 139, partitioning 139 = 100 + 39 and calculating 540 3 ∙ 100 5 ∙ 39 yield 45, a number not divisible by the difference in prices, 5 3. Likewise, replacing 540 by 541 without changing the rest of the data yields 51, another number not divisible by two. The Tutor problem data can be presented through a conceptual bond (Fig. 3.7) in which the apex holds the whole pyramidal structure in a sense that the money spent on 138 yards of cloth with the given costs for a yard may vary from the smallest sum (one yard of blue cloth, the rest – black), 416 3 ∙ 137 5 ∙ 1, to the largest sum (one yard of black cloth, the rest – blue), 688 3 ∙ 1 5 ∙ 137. In connection with the reference to the smallest and the largest sums, one can recall Euler’s explanation of the general significance of this issue: “since the fabric of the world is the most perfect and was established by the wisest Creator, nothing happens in this world in which some reason of maximum or minimum would not come to light” [cited in Pólya, 1954, p. 121]. Indeed, problems seeking the largest and the

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smallest values of a certain contextual characteristic have been “of interest not only in mathematics but also in everyday life where people often deal with questions of the best or worst and the least or most of a certain quality or quantity of behavior and of features of the social and physical world” [Luchins and Luchins, 1970, p. 301].

Fig. 3.7. The apex (yards bought) holds the problem structure.

Fig. 3.8. Repeating pairs results in pairs.

One can note that because the difference between the costs for a yard of each type of cloth is two, the sum of money spent should be a multiple of two. In addition, 138 is an even number and therefore, its additive partitions in two parts are either both even or odd. Furthermore, both prices for a meter are odd numbers. Noting that the product of two odd numbers is an odd number and the product of two numbers of different parity is an even number implies that a linear combination of the moneys spent on each type of cloth is always an even number. As shown in Fig.

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3.8, when repeating the combination of two bars one of which exceeds another by two units, whatever the number of repetitions, the resulting quantity consists of the pair of equal parts augmented by the pair of units. Decontextualizing the act of pairing confirms divisibility by two. That is, repeating pairs yields pairs. So, one has to find the maximum and the minimum values of the linear combination 3x + 5y for x  y  138 . The search can be reduced to exploring the function of one variable, say, f (x)  5 138  2x when 1 ≤ x ≤ 137. Because the function f(x) decreases monotonically, fmax  f (1)  688 and fmin  f (137)  416 . That is why (Fig. 3.7), any even sum of money in the range [416, 688] works for the apex 138 provided that the two bottom vertices don’t change. In fact, the bottom vertices may be changed to another pair of odd (or even) numbers, but this will affect the range for the vertex at the top. Likewise, any even number of yards of cloth varying in the range [110, 178] can be bought for the given prices and total sum (Fig. 3.9). Indeed, it follows from the 5 equation 5x + 3y = 540 that y  (108  x ) and therefore, in order to 3 make the difference 108 – x divisible by three, the largest x = 105 yields the smallest y = 5 whence x + y = 110 and the smallest x = 3 yields the largest y = 175 whence x + y = 178. This result can be confirmed by using a spreadsheet, which can be also used to generate a variety of problems similar to the one described in the above story [Tchekoff, 1970]. That is, formal reasoning can be enhanced by the use of technology that provides a TITE problem-posing environment informed by understanding the role of conceptual bonds, like those shown in Fig. 3.7 and Fig. 3.9.

Fig. 3.9. The apex (money paid) holds the problem structure.

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3.6 Revisiting mathematical problems to make them didactically coherent 3.6.1 From numerical to contextual coherence The use of technology supported by the framework of didactical coherence in problem posing (Chapter 2) enables mathematics educators to revisit mathematical problems used in the pre-digital era with the goal to enhance mathematical preparation of pre-college students. A new educational goal could be to modify such problems through an alternative formulation that makes problems didactically coherent and enables the use of technology in support of symbolic computations. In other words, by revisiting problems of the past, one can contribute to the development of TITE mathematics curriculum. To this end, consider the following problem used in the past in support of afterschool activities of secondary school students with special interest in mathematics [Sivashinsky, 1968]: Problem 3.6.1. Natural numbers are put in groups as follows: (1), (2, 3), (4, 5, 6), (7, 8, 9, 10), (11, 12, 13, 14, 15), ... . Find the sum of numbers in the 10th group. Find the sum of numbers in the nth group. This problem, seeking information, is numerically coherent as the set of natural numbers is closed under the operation of addition. That is, the sum to be found does exist regardless of the group’s rank. However, the problem does not have any context to be used to facilitate a transition from the first-order symbols to the second-order symbolism where generalization can take place. Just as “the written language of children develops ... [by] shifting from drawings of things to drawings of words” [Vygotsky, 1978, p. 115], the use of formal mathematical symbolism develops by ascribing quantitative meaning to concrete objects (used as frames of reference) that one can manipulate and/or visualize. With this in mind, Problem 3.6.1 was included in the course “Creative problem solving” taught by the author for K-12 teacher candidates to discuss place of its possible modifications within the regions of the Venn

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diagram of Fig. 2.6 (Chapter 2). As is, the problem belongs to region E – it possesses an obvious numerical coherence, has no context, and offers little grade appropriate pedagogy due to the absence of any frame of reference typically supported by the first-order symbols. In this course, the task for teacher candidates was to reformulate Problem 3.6.1 to allow for an exoteric context and engaging pedagogy, thereby, making it didactically coherent and, most importantly, accessible to a broader population of students whose interest in mathematics is in its infancy. To this end, one of the course participants posed Problem 3.6.2. A group of people is in a room together for some kind of meeting. Each person is expected to become part of a group. The first group will have one person, the second group will have two people, the third group will have three people, and so on. The person in the group one is assigned the number one. The persons in the group two are assigned the numbers two and three. The persons in the group three are assigned the numbers four, five, six, and so on with the remaining groups. Each person in each group is given a piece of candy according to their number. For example, person one gets one piece of candy, person two gets two pieces of candy, person three gets three pieces of candy, and so on. The person handing out the candy wants to put each group’s candy in a zip lock bag prior to handing it out and needs to know the total number of pieces of candy each group will get. Help this person to solve the problem. A classroom discussion of Problem 3.6.2 then ensued. It was concluded that whereas the problem does have conventional context appealing to students, its pedagogy does not offer any support system in dealing with finding the sum of consecutive natural numbers in a generalized situation (e.g., when such a sum begins with a number different from one). The class decided that the problem belongs to region A of the Venn diagram of Fig. 2.6 (Chapter 2) meaning that it is numerically and contextually coherent and lacks pedagogical coherence to be considered being didactically coherent. In order to facilitate an

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argument and motivate extended problem posing, a spreadsheet environment for calculating the number of candies in each bag was constructed by the author from where the sequence 1, 5, 15, 34, 65, 111, ... (column A in the spreadsheet of Fig. 3.10, see Appendix for programming details) was derived showing the number of candies in a bag the rank of which is located in the corresponding cell of (hidden) column B. In the context of Problem 3.6.1, the number in cell A11 is the sum of numbers (located in the region C11:L11), that belong to the 10th group, is equal to 505. Below it will be shown how the sequence of numbers in column A can be generalized to the n-th bag (alternatively, to the n-th group in Problem 3.6.1) using technology.

Fig. 3.10. A spreadsheet showing numerical coherence of Problems 3.6.1 and 3.6.2.

3.6.2 Towards pedagogical coherence The modeling data provided by the spreadsheet of Fig. 3.10 allowed for the formulation of Problem 3.6.3. There are 111 [Fig. 3.10, cell A7] candies in a bag prepared for one of the groups. What can be said about this group? This variation of Problem 3.6.2 is a TITE problem as its solution integrates argument and computation provided by technology. Problem 3.6.3, however, is not obviously numerically coherent. Its numerical coherence is obvious to a problem poser but not to a problem solver.

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This, however, is the major characteristic of technology-enabled problem posing – whereas problem posers (e.g., teachers) possess knowledge about their problem’s solvability, problem solvers (e.g., students) when encountering the problem not only don’t know the answer but are unsure whether it even exists. A problem solver may begin solving Problem 3.6.3 in a traditional way by denoting x to be the smallest number of candies a person gets from the bag with 111 candies prepared for n people. Carrying out the summation of n terms of the arithmetic series , 1, 2, … , 1 yields the equation 111, which is equivalent to the quadratic equation 2 1 222 0. In order for the last equation to have an integer solution, it is necessary for its discriminant, 888 , to be a square number. A computational task is to find 2 1 the smallest value of x under this condition. Once again, Wolfram Alpha can find that 31 888 43 . Therefore, 2 1 31 whence x = 16 and 6. That is, 111 candies from this bag can be distributed among six people through the following partition: 111 16 17 18 19 20 21. Note that whereas it is quite appropriate to deal with the expression 2 1 888 using technology, the very development of the quadratic equation was technology-immune and, thereby, Problem 3.6.3 is indeed a TITE problem. 3.7 From modeling data to a general formula using technology In order to obtain the general formula for the sequence 1, 5, 15, 34, 65, 111, ... , one can enter it into the input box of Wolfram Alpha. As a result, the expression (n3 + n)/2 is returned (Fig. 3.11) as the general answer to Problem 3.6.2/Problem 3.6.1. (Alternatively, one can use the OEIS® which, in addition to the cubic expression, provides multiple, purely mathematical, contexts for the sequence). Such an ease of making generalization through the affordance of the modern-day digital tools available on-line justifies the need for posing TITE problems in which argument and computation (both numeric and symbolic) go hand by hand to allow for the appropriate use of technology. One cannot ignore the existence and, most importantly, availability of these tools because artificially excluding technology from doing mathematics by students

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does not advance the preparation of the 21st century workforce. To a certain extent, not only Problem 3.6.3 but Problem 3.6.2 as well may be considered being a TITE problem: whereas the generalization phase can be outsourced to Wolfram Alpha or OEIS®, computing the number of candies in each bag can be carried out in the paper-and-pencil environment. In addition, the creation of the spreadsheet shown in Fig. 3.10 can become an agency for mathematical activities by teachers. Just as “the teaching [of a child] should be organized in such a way that reading and writing are necessary for something” [Vygotsky, 1978, p. 117, italics added], mathematical problem posing and solving as intellectual activities for teacher candidates should be given an applied flavor as much as possible.

Fig. 3.11. Using Wolfram Alpha in generalization.

3.8 Formulating and solving a didactically coherent problem Whereas bags of candies provide context for dealing with abstract nature of the strings of consecutive natural numbers (the second-order symbolism), this context does not allow one to manipulate and/or visualize concrete objects involved in order to simplify the summation of numbers. Thus, our goal is to design an appropriate problem-solving pedagogy that ultimately would lead to a didactically coherent problem. Such pedagogy can be grounded in recourse to geometry the images of which can be used as the first-order symbols. Towards this end, another teacher candidate formulated

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Problem 3.8.1. John is making towers from blocks according to the pattern shown in Fig. 3.12. How many blocks does he need to build the 10th tower in this pattern? How many blocks does he need to build the nth tower in this pattern? The formulation of Problem 3.8.1 was due to the teacher candidate’s experience with similar problems such as constructing towers out of matchsticks (e.g., [Abramovich and Brouwer, 2011]; [Fujii, 2008]; [New York State Education Department, 1996]). In addition, familiarity with the latter type of problems motivated her to offer an extension of Problem 3.8.1 when the towers are constructed out of matchsticks to allow for a different type of counting techniques to be considered. Such counting of the matchsticks and its generalization to an arbitrary number of towers provide another interesting example of a TITE problem.

Fig. 3.12. John’s towers built out of blocks.

In a mean time, by analyzing step-towers of Fig. 3.12, one can see that each such tower comprises components, the heights of which are consecutive natural numbers. Now the sum of numbers in each group (Problem 3.6.1) or the number of candies in each bag (Problem 3.6.2) coincide with the number of blocks used to build the corresponding tower. One strategy to find the number of blocks in a tower of this

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structure is to augment it to have a rectangle which area is easy to find (Fig. 3.13). While a step-tower represents geometrically consecutive natural numbers the smallest of which (beginning from the second tower) is different from one, the augmentation always represents consecutive natural numbers the smallest of which is equal to one. Thus, the number of auxiliary blocks can be found without much difficulty. In each tower, its rank in the set of towers is the width. The height can be found in a number of ways. One way is to associate the height with the largest number in each group of Problem 3.6.1. Thus, the sequence of numbers 1, 3, 6, 10, 15, ... results. Once again, using Wolfram Alpha in identifying this sequence, yields a familiar expression, . Consequently, the ∙

number of blocks in the n-th tower is equal to

. At

the same time, the number of blocks augmenting the n-th tower to a rectangular shape equals to the height of the (n – 1)-th tower, that . Therefore, the number of blocks in the n-th tower gives the is, sum of numbers in the n-th group as follows: . One can say that Problem 3.8.1 is a TITE problem which is didactically coherent for it is easily computable, its frame of reference provides the first-order symbols in the form of towers, which, in turn, provide enjoyable context. The use of the first-order symbols facilitates transition to the second-order symbolism (a TI part) the correctness of which is verified through the use of technology (another TE part). The solution was supported by a ready-made spreadsheet that generated the sequences 1, 5, 15, 34, 65, 111, ... and 1, 3, 6, 10, 15, ... the general form of which was found using Wolfram Alpha. However, the intricacy of human-computer interaction is that a skill of feeding a number sequence into Wolfram Alpha should not be taken for granted as examples of using Maple [Char, Geddes, Gonnet, Leong, Monagan and Watt, 1991] to do mathematical induction proof will demonstrate in the next section.

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Fig. 3.13. Augmenting towers to facilitate counting of blocks.

3.9 Maple-based mathematical induction proof Another computer program which can be used in dealing with TITE problems is Maple. Without sophisticated programming, the software is capable of carrying out rather complex symbolic computations, yet the entire process has to be controlled by a user equipped with skills far beyond correct button pushing. In this section, several problems stemming from the previous sections and associated with the use of mathematical induction proof will be discussed. To begin, consider the expression which was shown (section 3.2) representing the sum of the first n natural numbers. That is, the formula 1 holds true for any

∈

2

3

⋯

. In particular, this implies that the product

1 is divisible by two for any ∈

. Therefore, one can pose

Problem 3.9.1. Prove by the method of mathematical induction that 1 is divisible by two for any ∈ . According to Pólya [1954], the main idea of this method is to “test the transition from n to n + 1” (p. 111). In the context of Maple, assuming that 1 is divisible by two and showing that the same is correct for the difference 1 implies that 1 is divisible by two. Maple yields the relation 1 2 2, the right-hand side of which, regardless of n, is obviously a multiple of two.

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Consider another example. The spreadsheet of Fig. 3.10 has generated in column E the sequence 6, 9, 13, 18, 24, 31,... . One can pose a problem of finding the sum of the first 100 terms of this sequence and, firstly, enter this sequence into the input box of Wolfram Alpha. The program responds with the nth partial sum formula (Fig. 3.14) which, in turn, allows for finding that the sum of the first 100 terms of the sequence is equal to 177150 (Fig. 3.15). Similar problems can be formulated by using number sequences generated in other columns of the spreadsheet of Fig. 3.10. Furthermore, as shown in Fig. 3.16, the quest “Find the sum of (1/2)*(i^2+3i+8) from i=1 to i=n” yields the partial sum formula in the form 6 29 . The ability of altering the formulation of a problem in order to make it solvable by software is an indication of one’s custodianship of knowledge of mathematical content and the appropriate use of technology.

Fig. 3.14. Wolfram Alpha in finding the general term of the sequence 6, 9, 13, 18, 24, 31, ….

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Fig. 3.15. Wolfram Alpha in finding the sum of the first 100 terms of the sequence 3 8 .

Fig. 3.16. Wolfram Alpha in finding a partial sum.

The expression

6

29

was found as a partial sum of an

integer sequence. Although the expression looks like a fraction, it produces integers only and, thereby, it has to be divisible by six for any ∈ . This observation can motivate posing Problem 3.9.2. Prove by the method of mathematical induction that 6 29 is divisible by six for any ∈ . This time, testing the transition from n to n + 1 cannot be completed by Maple in one induction step. Assuming that 6 29 is divisible by six yields, as shown in Fig. 3.17, that 1 3 15 36 – a trinomial not obviously divisible by six. Therefore, Maple has to be used again to prove this divisibility by mathematical induction. To this end, assuming that the expression 3

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15 1

36 is divisible by six, one has to show that the difference is divisible by six as well. Maple shows the difference 1 6 18. Recognizing that 6 18 is an obvious multiple of six completes the proof of divisibility by six of the trinomial 6 29 .

Fig. 3.17. Testing divisibility of

6

29 by 6 in two steps.

Likewise, by demonstrating in the context of Wolfram Alpha that the n-th partial sum of the series 10 + 14 + 19 + 25 + 32 + 40 + … (see column F in the spreadsheet of Fig. 3.10) is equal to the cubic trinomial , one can prove using the Maple-based method of mathematical induction that the trinomial 9 50 is divisible by six for any ∈ . Note that such use of Maple demonstrates how one can deal with appropriately posed TITE problems by outsourcing symbolic computations to the software. This novel way of using technology should not be taken to mean that human-computer interaction is always straightforward, even within an apparently identical mathematical context. Indeed, unlike the case of Problem 3.9.1 where proof required a single induction step, an apparently similar case of Problem 3.9.2 required two induction steps. Such intricacy of the Maplebased mathematical induction proof within a seemingly same class of TITE problems indicate the importance of informed assistance by a ‘more knowledgeable other’ in the problem-posing and problem-solving contexts of the digital era.

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Although transformations required for demonstrating the transition from n to n + 1 in the above examples are not really challenging and, perhaps, should belong to the (secondary mathematics) teacher candidates’ mathematical machinery tool kit, the following three perspectives on the use of technology in mathematics teaching appear to be relevant as the conclusion to this section. First, in the digital era, traditional paper-and-pencil techniques (e.g., algebraic transformations of a mathematical induction proof) can give way to new techniques made possible by the appropriate use of technology [Drijvers, 2000]. Second, such use of technology can be seen through the lenses of the theory of instrumental genesis (developed mostly by French researchers) and rooted in Vygotsky’s [1978] seminal idea of mediation by tools (both material and psychological), when new pedagogical ideas about the appropriation of a digital tool in support of a particular content involving problem posing and solving take shape [Guin and Trouche, 1999]. In terms of this theory, technology-immune part of tool-mediated problem solving may be construed as “instrumentation” – a process through which a subject (i.e., a user of a digital tool) develops intellectually; its technology-enabled part may be construed as “instumentalization” – a process through which an artifact (i.e., a digital tool) broadens the realm of utilization [Lonchamp, 2012, p. 216]. Finally, due to the advances in computer-supported learning, as Langtangen and Tveito [2001] put it, “Much of the current focus on algebraically challenging, lengthy, errorprone paper and pencil work can be significantly reduced. The reason for such an evolution is that the computer is simply much better than humans on any theoretically phrased well-defined repetitive operation” (pp. 811812). 3.10 Summary This chapter, starting from a brief history of the educational uses of computers in the United States, revealed challenges that technological advances created for mathematical education. One such challenge is that many traditionally formulated problems can be solved by a computer with almost no intellectual contribution by a student. This suggested the need for reformulating such problems; in other words, for creating their appropriate modifications in order to make the completion of problems

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immune from an automatic application of software. At the same time, the use of technology must be a critical part of solving a problem enabling its, otherwise inaccessible, solution. Towards the end of such a dual perspective on the use of technology, the chapter provided examples of activities when problem reformulation entails looking at the past as a way of developing new teaching ideas. The chapter reflected on the author’s work mainly with secondary mathematics teacher candidates. It was demonstrated that the skill of posing modifications of traditional problems requires robust conceptual understanding of a problem situation. In particular, the use of technology in carrying out a mathematical induction proof was suggested. In the next chapter, mathematical problem posing will be presented as a means of linking procedural competence and conceptual understanding.

b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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Chapter 4

Linking Algorithmic Thinking and Conceptual Knowledge through Problem Posing 4.1 On the hierarchy of two types of knowledge The National Council of Teachers of Mathematics [2000] has strong belief that school mathematics curriculum must be “offering students opportunities to learn important mathematical concepts and procedures with understanding” (p. 3). Likewise, in the Common Core State Standards [2010] mathematical understanding is referred to as “the ability to justify … why a particular mathematical statement is true or where a mathematical rule comes from” (p. 4, italics in the original). Whereas these documents, commonly seen as the hallmark of the 21st century mathematics education endeavors in North America, do not attempt arranging mathematical understanding and procedural skills in the order of importance, there has been continuous debate regarding the hierarchy of the two types of mathematical knowledge. The goal of this chapter20 is to contribute to this debate and use the context of problem posing in order to highlight the dichotomy of perspectives concerning the relationship between conceptual understanding and algorithmic (procedural) thinking. The first mathematics education publications on the topic of what develops first – procedural or conceptual – appeared in the early 1980s. At the preschool level, Gelman and Meck [1983], based on their study of counting skills by young children, argued that basic principles of counting have to be developed first in order to use counting as a skill. Nonetheless, the development of the basic principles does not imply that one has full conceptual understanding of ideas associated with counting at the level of abstraction (e.g., comprehending the infinity of the set of number names, something that implies that counting may never stop). Following

20

This chapter is an extended version of the author’s paper [Abramovich, 2015b]. 85

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the studies by psycholinguists (e.g., [Clark and Clark, 1977]), Gelman and Meck [1983] made a distinction “between implicit and explicit knowledge of counting principles [similarly to how] young children are granted implicit knowledge of language rules well before they are said to have explicit knowledge of their grammar” (p. 344). Indeed, just as a second grader, when measuring distance from a mark on the floor to the wall, intuitively puts a measuring tape perpendicular to the wall, an observation made by the author in the field [Abramovich, 2017, p. 38], “the child conjugates and declines correctly but without realizing it” [Vygotsky, 1987, p. 109]. Freudenthal [1983], one of the leading mathematics educators of the second part of the 20th century, seemed to favor students’ conceptual understanding versus procedural performance when saying that “sources of insight can be clogged by automatisms” (p. 469); thereby, warning against nondiscriminatory use of algorithmic procedures without cognizant thinking. Many examples in support of the last quote are presented in Chapter 6 through the discussion of Einstellung effect, something that stands in the way of using productive thinking (alternatively, insight) in problem solving. Several authors, Haapasalo and Kadijevich [2000] among them, found that in the early mathematics education procedural skills typically enable concept development. For example, by representing integers as a sum of two integers, one can develop conceptual understanding that only even numbers can be represented as a sum of two equal numbers. Later, this understanding can be extended to geometry; namely, only even numbers may serve as perimeters of integer sided rectangles, and the smaller the difference between the side lengths, the larger the number of unit squares the rectangle comprises. Nonetheless, automatically counting points on a circle may be used as a counterexample to the universal acceptance of the primordial nature of procedural vs. conceptual without a rather obvious codicil that no procedural skill is possible in the absence of some basic (or implicit, as Gelman and Meck [1983] called it) conceptual understanding of the skill. Even counting, either points on a circle or unit squares within a rectangle, requires one to start with attaching the name one to the first counted object and then using each number name only once following their order.

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Technology has brought about new didactic approaches to understanding interplay between the procedural and the conceptual. At the dawn of the digital era, Nesher [1986] argued that “what remains for us to do [when using technology] is to consider which algorithms we will want to use to free ourselves from thinking, and which can be best used in order to further our thinking and understanding” (p. 8). It is not clear, however, how something can be done with technology without thinking. Even in the simplest case of using a calculator to add two three-digit numbers, one may not be entirely free from thinking as pushing a wrong button on a calculator can give an erroneous result which cannot be recognized in the absence of thinking. For example, on a calculator the plus button and the division button look almost the same and, in the author’s experience observing second-grade mathematics lessons with the use of a calculator, pupils frequently confuse the buttons, especially in the darkness of the classroom. So, when adding two (positive) threedigit numbers, the plus button should yield another number with at least three digits. Yet pushing the division button, most likely, would not bring about even a whole number which is the only type of a number familiar to young children. Making such a distinction is not possible without thinking. What Nesher [1986], perhaps, had in mind is the need for a basic conceptual understanding of an operation rather than the absence of thinking when carrying out this operation using technology. This argument highlights the duality between the use of technology with minimal thinking and the importance of thinking in support of technological developments. In the late 1980s – early 1990s, as technology has become more sophisticated, Kaput [1992] called for disciplined inquiry into “the relation between procedural and conceptual knowledge, especially when the exercise of procedural knowledge is supplanted by (rather than supplemented by) machines” (p. 549). As a way of addressing this call, Peschek and Schneider [2001] put forth the notion of outsourcing algorithmic skills to new digital technology such as computer algebra system (CAS), something that, nonetheless “presumes thorough basic knowledge in mathematics; … the willingness and the ability … to be precise when formulating one’s own questions … in a form which can be interpreted by CAS” (p. 17). As noted by Kadijevich [2002], the use of CAS has the potential to facilitate links between

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procedural skills and conceptual understanding because even relatively unsophisticated symbolic computations “require the user to think conceptually before a procedure is used” (p. 72). Indeed, as discussed in Chapter 3 (section 3.1), without conceptual understanding of what the notation INT really means, it is quite possible for a user of Wolfram Alpha to overlook that the program interprets INT as antiderivative rather than the greatest integer function. That is, human-computer interaction in the context of outsourcing symbolic calculations to a digital tool does require users’ conceptual understanding of the procedures involved. An interesting contribution to the debate on the hierarchy of the two types of knowledge was to suggest “that conceptual and procedural knowledge [in mathematics] develop iteratively, with increases in one type of knowledge leading to increases in the other type of knowledge, which trigger new increases in the first” [Rittle-Johnson, Siegler and Alibali, 2001, p. 346]. These authors defined the notion of “problem representation as the internal depiction or re-creation of a problem” (ibid, p. 348, italics added) formed by a problem solver each time the problem is solved. That is, solution of a problem precedes its re-enacting in a problem-solving setting. One’s ability to re-enact a problem brings to mind the notion of reflective inquiry [Dewey, 1933] – a way of learning when experience leads to the growth of knowledge and, vice versa, knowledge makes experience profound. In mathematics, reflective inquiry can be interpreted as an act of posing a new problem after a similar problem is solved. So, considering the relationship between procedural skill and conceptual knowledge as an iterative alliance leads to the interpretation of problem posing as a recurrent reflection on a solved problem through the (potentially) never-ending cycle “solvereflect-pose”. While the “solve-reflect-pose” cycle is a typical mathematical technique, one way to pursue it in mathematics teacher education with the minimal contribution of a (‘more knowledgeable’) instructor is through the process of reciprocal problem posing. This process may involve a pair (or two groups) of teacher candidates who, using a basic problem (perhaps offered by the instructor) as a springboard, start posing different extensions of the problem for each other. By experiencing reciprocity in posing problems, teacher candidates can learn that one of

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the main difficulties involved in this process is finding the right balance between the challenge and the frustration, sentiments commonly associated with solving problems. Even Hilbert [1902] advised mathematicians, “a mathematical problem should be difficult … yet not completely inaccessible” (p. 438). Nowadays, the cycle “solve-reflectpose” can be enhanced procedurally through the use of computational environments specifically designed by the instructor for posing problems of a certain type (e.g., finding all ways of changing a quarter into dimes, nickels, and pennies) and expanded conceptually through paying attention to the notion of didactic coherence of a problem (e.g., thinking about students’ familiarity with the names of the coins or about ways of reducing the multiplicity of answers). Also, the cycle can be sustained through the instructor’s competent guidance in engaging teacher candidates into the discussion of mathematical ideas leading to the development of the taste of problem posing (and, consequently, problem solving) by asking (and then answering) second-order questions (i.e., questions requesting explanation). Within this kind of classroom discourse, one can begin appreciating how a slight modification of a simple question or a search for an alternative procedure quite unexpectedly may become a source of conceptual developments in mathematics. As a response to a first-order question (i.e., a question seeking information), one asks a second-order question trying to understand why a procedure works, what its limitations are, and whether and how it can be applied in a similar situation. The next section shows how teacher candidates’ acceptance of a non-traditional response to a first-order question can uncover hidden creativity of young children and facilitate the transition to posing second-order questions. 4.2 A simple question leads to revealing hidden creativity What is creativity? Educators see creativity as “one of the essential 21st century skills … vital to individual and organizational success” [Beghetto, Kaufman and Baer, 2015, p. 1]. Teachers’ ability to recognize creativity of their students that may be hidden behind their immature classroom performance, is critical for successful teaching and productive leaning. If students’ hidden creativity is not acknowledged and supported by a teacher, it would most likely remain dormant if not vanish. The

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following story, drawn from a second-grade classroom, supports the notion that teachers are the major custodians of unfolding creative potential of young children. An elementary teacher candidate, during his practicum (pre-student teaching), asked a student to construct all possible rectangles out of eight square tiles. This is a typical activity aimed at preparing children for learning multiplication – each rectangle represents a multiplication fact (the second-order symbolism) expressed through the first-order symbols. The teacher candidate’s expectation was to see two rectangles as shown in Fig. 4.1. Defying the expectation of the teacher candidate, the student, in addition to these two rectangles, offered an example of a 3x3 square (after all, square is a special case of rectangle) with a missing tile in the middle of the square (Fig. 4.2).

Fig. 4.1. Eight tiles – two rectangles with no windows.

Fig. 4.2. A rectangle (square) with a square window.

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Being taught in his own mathematics education course about the need to pay attention to children’s non-conventional ideas that do not necessarily meet traditional expectations and very often are indicative of hidden creativity, the teacher candidate accepted this solution with a praise. Later, he shared his experience with the author, his mathematics education professor, who, in turn, recognized hidden mathematical meaning of the second grader’s example of a square with a window offered in addition to two (windowless) rectangles. This discovery, apparently made by serendipity, was due to two factors – the number of tiles given to the student, eight, and his emerging hidden creativity due to the first factor. A large number of teaching ideas that stem from this example can be revealed. For brevity, only the following three questions will be mentioned. Question 1. Can such “rectangle” (square) with a window be constructed out of nine tiles? Question 2. If not, what is the smallest number of tiles, greater than eight, out of which such construction is possible? Question 3. What is the smallest number of tiles out of which more than one rectangle with a window can be constructed? The last question requires clarification. For example, out of 11 tiles one can construct a 3x4 rectangle with a 1x1 window located in two different places within the rectangle. But what if more than one rectangle has to be constructed? It appears that 14 is the smallest number of tiles out of which two such rectangles can be constructed as 14 = 3x5 – 1x1 = 4x4 – 2x1. This information prompts another second-order question: Is this the case for any even number of tiles greater than or equal to 14? Eventually, the discussion stemming from hidden creativity of the second grader may prompt the following query: Can search for such rectangles be computerized? This is where mathematics and technology meet to enable, in the spirit of Montessori [1917] who, long before the digital era, emphasized the importance of “the recognition of new phenomena, their reproduction and utilization” (p. 73) for education, linking problem posing to mathematical experiment. Through the engagement into experimentally supported problem posing, one develops conceptual understanding of mathematics, something that can be interpreted in terms of turning a doing into an undoing [Mason, 2000]; that is, using a solved

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problem as a means of asking a (second-order) question about conditions that made it solvable. In analyzing a problem posed, one has to make certain that it is numerically coherent and decide under what conditions its numeric data provides a solution. 4.3 Two levels of conceptual understanding In parallel with two types of semiotic mediation frequently employed by mathematics educators – the first- and second-order questions and the first- and second-order symbols used, respectively, as a frame of reference in support of contextualization and notation for decontextualization, one can distinguish between two types of conceptual understanding – basic conceptual understanding and advanced conceptual understanding. Basic conceptual understanding allowed the second grader to offer a non-traditional rectangle. Yet the second-order questions posed above (Questions 1–3) in response to the emergence of such a rectangle necessitate advanced conceptual understanding. Mathematical problem solving requires some basic level of conceptual understanding of the situation in terms of the first-order symbols. The second-order symbolism calls for advanced conceptual understanding. For example, by knowing that the number of objects in a set is one more than in another set, one may decontextualize from the concreteness of this information and describe the two sets through the symbols n and n + 1. For that, one has to see the symbol n as a variable (unknown) quantity and n + 1 as a process of increasing this quantity by one. In other words, de-contextualization requires one to see the first-order symbols as a concept and to think about the use of the second-order symbolism as a process [Tall et al., 2001]. In the case of the rectangle with a missing tile (Fig. 4.2), one can see the square numbers 32 and 12 as well as the products 4x3 (i.e., 3 + 3 + 3 + 3) and 4x1 (i.e., 1 + 1 + 1 + 1) as the concepts of area and perimeter, respectively. At the same time, the differences 32 – 12 and 4x3 – 4x1 can be seen as the process of comparing areas and perimeters of the squares. It turns out that 32 – 12 = 4x3 – 4x1. Surprise: the difference between areas is equal, numerically, to the difference between perimeters. Replacing the numbers 3 and 1 by the variables n and m, this process is described by the equation n2 – m2 = 4(n – m) being equivalent (as n – m

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0; otherwise, there is no square) to the equation n + m = 4 with the single solution n = 3 and m = 1 (represented through the first-order symbols in Fig. 4.2). In other words, the rectangle with a window constructed by the second grader is unique in terms of the process carried out upon the areas and perimeters being considered as concepts. One can see that basic conceptual understanding is necessary in order to activate problem solving, e.g., constructing rectangles using tiles. Advanced conceptual understanding is necessary to reflect on problem solving through posing new questions, like those mentioned in section 4.2. In the words of Walter and Brown [1971], problem solvers “should have some machinery available to define and attack problems” (p. 402, italics in the original). Nonetheless, in many cases, basic conceptual understanding is not sufficient for either solving a problem or finding an efficient solution. Indeed, even a simply looking problem, like constructing rectangle out of eight square tiles, may have “unsuspected depth” (ibid, p. 402). Without advanced conceptual understanding, it was not possible to recognize hidden creativity of the second grader who did not suspect that his example is conceptually rich, something that can be revealed with the help of ‘a more knowledgeable other’ if not right away in the classroom, then, at least, through a follow up reflection on the lesson. In any case, such practical experience can lead to the growth of knowledge which, in turn, brings an intelligent insight into the experience and, through the lenses of advanced conceptual understanding, making it profound. 4.4 Solving a problem seeking information As another illustration of the interplay between basic conceptual understanding and advanced conceptual understanding, consider. Problem 4.4.1. The sum of three consecutive natural numbers is equal to 81. Find the numbers. This problem seeks information about a certain type of integers, three of which have a given sum. Its algebraic solution requires some basic conceptual understanding; namely, that any three consecutive integers form an arithmetic progression with difference one. Thus, the three

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integers can be written as n, n + 1, and n + 2 from where the equation 1 2 81, then the value 26, and, finally, the triple of integers (26, 27, 28) follow. The problem, however, can be solved differently, in a purely arithmetical way, without any explicit use of algebra (cf. the problem in a classic story Tutor discussed in Chapter 3, section 3.5). Instead, one needs to possess advanced conceptual understanding of the problem structure. This understanding can be developed through requesting explanation as to why the sum of three consecutive terms of any integer arithmetic sequence (including natural, even, and odd number sequences) is a multiple of three. To this end, one can draw a diagram as a way of answering the second-order question by using the first-order symbols. Consider the diagrams of Fig. 4.3 and Fig. 4.4. They show, respectively, that any 3-step staircase representing the sums of three consecutive natural numbers and, in general, integers in arithmetic progression can be rearranged into a 3-layer rectangular podium regardless of the length n of the upper step and, therefore, without loss of generality, n may be replaced by 1. Indeed, when the bottom right step is moved up as shown in Fig. 4.3 and in Fig. 4.4, the length of the upper level is irrelevant for carrying out this move and thus it may be assumed having the smallest possible integer length. That is, one can perceive abstract symbol n as a concrete (particular) concept embedded into the context of straightening out a 3-step staircase of a special type. This, of course, requires the ability to contextualize by probing into the referents provided by the first-order symbols. In that way, advanced conceptual understanding developed through the interplay between the second-order questions and the first-order symbols makes the corresponding solution purely arithmetical: divide 3 into 81 to get 27 which then has to be diminished and increased by one yielding 26 and 28, respectively. That is, on the one hand, in order to answer a question that seeks information about three consecutive integers with a given sum (the first-order question), one transforms basic conceptual understanding of how these integers are related into an algebraic equation which yields a solution through a well-defined algorithm. Yet numeric information so obtained without reflective inquiry into why the algorithm works does not necessarily help one to

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develop advanced conceptual understanding. On the other hand, by answering a question requesting explanation of the process through which numeric information was obtained, brings both an efficient numerical algorithm and advanced conceptual understanding. The described interplay among arithmetic, algebra (“generalized arithmetic” [Philipp and Schappelle, 1999]), basic conceptual understanding, and advanced conceptual understanding is quite significant: whereas formal algebraic solution required basic conceptual understanding, purely arithmetical solution required advanced conceptual understanding.

Fig. 4.3. Turning a 3-step staircase into a 3

1 -rectangle.

Fig. 4.4. Turning a 3-step staircase into a 3

-rectangle.

4.5 Problem posing leads to conceptual knowledge and collateral learning Advanced conceptual understanding can serve another purpose: it is necessary for posing a similar problem through reflecting on the one previously solved. Indeed, basic conceptual understanding may not be enough to pose a numerically coherent problem by altering data without reflecting on the data through answering the (self-posed) second-order questions. For example, replacing 81 by 80 yields no solution. Advanced conceptual understanding suggests that in the context of consecutive

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natural numbers one only needs to substitute 81 by any multiple of three greater than 5, a property that the sum of any three consecutive integers possesses, to have a similar problem which is solvable in integers. But 81 is not just a multiple of three, it is an odd multiple of three. So, whereas the sum of three consecutive odd numbers is also an odd multiple of three, the sum of three consecutive even numbers is an even multiple of three. In that way, in posing Problem 4.4.1, the quadruple of data source that includes the sum of numbers in arithmetic progression, their quantity, the difference, and the type of numbers (determined by the first number) were selected to be in a conceptual bond (similar to the one used in Chapter 3, section 3.5). Fig. 4.5 (where the letters N, O, and E stand, respectively, for natural, odd, and even numbers) shows how one can go beyond basic conceptual understanding and through the abovementioned cycle “solve-reflect-pose” develop advanced conceptual understanding. The far-left diagram of Fig. 4.5 shows Problem 4.4.1, the next diagram shows that in the case of three integers being consecutive odd numbers (the difference d = 2) the sum 81 may be preserved, but in the case of consecutive even numbers the sum has to be replaced by an even multiple of three. At the same time, unless a counterexample is provided, an alteration of data leading to a solvable problem may be accidental and it is not due to advanced conceptual understanding.

Fig. 4.5. The cycle “solve-reflect-pose” stemming from Problem 4.4.1.

Fig. 4.6. Variation of the number of terms yields multiple solutions.

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One of the common threads permeating the entire school mathematics curriculum (and consequently, course work for the future teachers of mathematics) is the representation of numbers as sums of other numbers. For example, integers (not all though) may be represented through the sums of consecutive natural numbers; squares of integers – through the sums of odd, triangular, and square numbers; unit fractions – through the sums of like fractions; integers – through irrational numbers; real numbers – through complex numbers; and so on. All such representations, although emerging from the algorithms of different levels of complexity, are often treated as the elements of procedural knowledge. What many teacher candidates (especially elementary ones, in the author’s experience) don’t appreciate is the significance of the fact that such representations may not be unique or do not exist at all. In the context of ideas brought to light by Problem 4.4.1, the following question can be asked: Why do the integers 13, 15, and 16 have, respectively, one, three, and zero representations as a sum of consecutive natural numbers? It is through asking such second-order questions that advanced conceptual understanding of seemingly insignificant (through the lenses of basic conceptual understanding) mathematical situations can be developed. In many cases, the first-order symbols can be used to facilitate such conceptual development in mathematics teacher candidates, something that they can later use in their own teaching. Skills in asking conceptual questions about mundane procedures, which often seen as the main mathematical experience of schoolchildren, are useful not only because of their relevance to the whole precollege curriculum, but, better still, such skills are important for preparing schoolchildren to study mathematics at the tertiary level. With this in mind, note that allowing the number of consecutive integers to vary (the far-right diagram of Fig. 4.5) leads to a problem with several correct answers all of which are shown in Fig. 4.6. Similarly to the case of the merchant problem, discussed in Chapter 3 (section 3.5), the tetrahedron-like diagrams of Fig. 4.5 and Fig. 4.6 show how the apex of the conceptual bond, the sum of numbers, holds the whole structure. Note that for any integer apex (a sum of an arithmetic series) one can always find three vertexes of the base enabling a numerically coherent problem. However, in the case when the sum is a prime number, only a

98

Integrating Computers and Problem Posing in Mathematics Teacher Education

trivial solution (partition into two consecutive integers) exists. The conceptualization of the last statement can be facilitated by the formula ,

(4.1)

an algorithm for finding the sum S of n terms of the arithmetic series with the first term x and difference d. Formula (4.1), in the case when S is a prime number, yields n = 2 whence . The last relation, obtained as a reflection on a formula defining a computational algorithm, shows that d can only be an odd number. For example, when S = 79 one possible solution is the representation 39 + 40 = 79 related to d = 1. In particular, it follows that a prime number cannot be partitioned into three or more integers in arithmetic progression. In connection with the last statement, a mathematical result due to Dirichlet21 is worth mentioning. Namely, if d ≥ 2 and x ≠ 0 are relatively prime, then there are infinitely many prime numbers among the terms of the sequence , n = 0, 1, 2, … . One can see that whereas by adding three or more integers in arithmetic progression it is impossible to reach a prime number, there are arithmetic sequences that contain infinitely many primes. This is an example of how by appreciating the tenet “Perhaps the greatest of all pedagogical fallacies is the notion that a person learns only the particular thing he is studying at the time” [Dewey, 1938, p. 49] one can become aware of mathematical results that belong to purely conceptual knowledge22 through collateral learning. This kind of learning is especially important within a course for prospective teachers of secondary mathematics enabling their familiarity with advanced subject matter context without the need to have the formal understanding of the context [Stewart, 1990]. Towards this end, one develops connections among different mathematical concepts through the cycle “solve-reflect-pose” that includes the recurrent integration of procedural and conceptual knowledge. The above example of using formula (4.1), an element of mathematicians’ tool kit, in developing 21 22

Peter Gustav Lejeune Dirichlet (1805-1859) – the outstanding German mathematician. Indeed, Dirichlet provided no formula (algorithm) for generating prime numbers.

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99

advanced conceptual understanding through problem posing may explain why, as it was noted by McCallum [2015], many mathematicians believe that practicing algorithmic skills (including the creation and the use of formulas) can lead to conceptual understanding. Whereas the creation of a formula, in many cases representing just a counting algorithm, can be achieved through the reflection on the formula by using basic conceptual understanding alone, advanced conceptual understanding emerges and further matures through collateral learning. Often, one begins the cycle “solve-reflect-pose” with only basic conceptual understanding and gradually develops advanced conceptual understanding, which could be used as basic conceptual understanding at the next iteration of the interplay between the procedural and the conceptual. In that way, problem posing enables a multi-level transition from basic conceptual understanding to advanced conceptual understanding. 4.6 Using conceptual bond in posing problems with technology One can integrate the conceptual bond structured by formula (4.1) and an electronic spreadsheet in order to pose problems similar to Problem 4.4.1 through a computational experiment. This problem-posing experiment would not be possible without a reflective inquiry into the solved problem that seeks explanation of the success of problem solving. In particular, numeric data generated by the spreadsheet of Fig. 4.7 allows one to formulate Problem 4.6.1. The sum of five integers forming an arithmetic sequence with difference seven is equal to 85. Find these integers. Possessing advanced conceptual understanding about a sum of an odd number of terms in arithmetic progression being a multiple of that number, one can divide 5 into 85 to get 17 – the average of the five numbers, from where other four numbers result: 3, 10, 24, and 31. A multitude of like problems can be posed using the spreadsheet shown in Fig. 4.7, where the entries of cells A3 (controlled by a slider, thus enabling variation), C2, and D2 correspond to the vertexes of the base of

100 Integrating Computers and Problem Posing in Mathematics Teacher Education

the conceptual bond and the entry of cell B2, the sum, is computed through formula (4.1).

Fig. 4.7. Spreadsheet as a problem generator.

The process of reformulation of a mathematical problem aimed at posing like problems, can be seen as a reflection on a solved problem in search of new ones. This perspective enriches both procedural skills and conceptual understanding. That is, the usefulness of reformulation (alternatively, problem posing) for the learning of mathematics is in a possibility of linking two types of knowledge. The spreadsheet can also be used to pose more complicated problems similar to the one represented through the tetrahedrons of Fig. 4.6. This requires certain level of procedural skills. To this end, setting d = 1 and x + n – 1 = m in formula (4.1) yields 1

2

(4.2)

where x and m are, respectively, the first and the last terms of the sequence of consecutive natural numbers with the sum S. Formula (4.2) shows that any representation of its right-hand side as a product of two integers of different parity defines a distinct tetrahedron with the apex S. For example, when S = 81 there are four such products with the smaller factor representing the number of terms, namely, 162

81 ∙ 2

54 ∙ 3

27 ∙ 6

18 ∙ 9.

Note that each of the products includes an odd divisor of 81 except 1. That is, the number of such divisors (which, as shown in Fig. 4.8, can be

Linking Algorithmic Thinking and Conceptual Knowledge through Problem Posing 101

found in the context of Wolfram Alpha) is equal to the number of ways an integer can be represented as a sum of consecutive natural numbers. Once again, the appropriate reflection on a formula, which was due to the application of a procedural skill, develops advanced conceptual understanding of a rather complex mathematical structure referred to by Pólya [1965] as a trapezoidal representation of an integer (see also [Abramovich, 2017, Chapter 6]). In the case of 81, all trapezoidal representations are shown beneath the tetrahedrons of Fig. 4.6. Likewise, one can use conceptual knowledge developed through the algebra-free solution of the merchant problem (Chapter 3, section 3.5) in order to use a spreadsheet in posing similar problems.

Fig. 4.8. Using Wolfram Alpha to find the divisors of 81.

Finally, as a demonstration of Dirichlet’s theorem mentioned in section 4.5, one can also use this spreadsheet to generate arithmetic sequences of a sufficient length with the (relatively prime) first term and difference and check their terms for primality. The 946th term of the sequence with x = 2 and d = 3 is equal to 2837 (Fig. 4.9, cell A948). Once again, one can use Wolfram Alpha to check this number for primality (Fig. 4.10), by asking for its divisors. By showing exactly two divisors of 2837, one and itself, the program confirms that 2837 is a prime number. This is just one of many examples of using integrated spreadsheets [Abramovich, 2016a] as a paradigm of Type II technology application [Maddux, 1984] in exploring mathematical ideas.

102 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 4.9. Is 2837 a prime number?

Fig. 4.10. Wolfram Alpha in deciding the primality of a number.

4.7 Summary This chapter shared the author’s contribution to a debate that has been taking place among mathematicians and mathematics educators regarding the order of importance between two types of knowledge – procedural and conceptual. Instead of ranking the two types, it was suggested that mathematical problem posing can be used as a bridge between the procedural and the conceptual. Two types of questions were discussed – the first-order questions seeking information (the source of which is mostly procedural) and the second-order questions requesting conceptual explanation of algorithmically obtained information. The notion of hidden creativity that can be observed in a second-grade

Linking Algorithmic Thinking and Conceptual Knowledge through Problem Posing 103

classroom was discussed and two levels of conceptual understanding, basic and advanced, were suggested. It was shown how advanced conceptual understanding emerges from interplay between the secondorder questions and what Vygotsky [1978] had referred to as the firstorder symbols. In other words, it was shown how concrete materials (the first-order symbols), when used in the classroom to spark basic conceptual understanding, are conducive to the development of advanced conceptual understanding through problem posing. The notion of conceptual bond, something that provides a problem with structure needed for a proper conceptual understanding, was shown serving as a vehicle for posing problems in the technological paradigm. The next chapter is devoted to the use of computer graphing software as a problem-posing and problem-solving environment for algebraic equations with parameters. It will be shown how explorations of that kind, being rudimental to real problems arising in STEM disciplines, provide secondary teacher candidates with valuable research-like experiences needed for successful teaching of mathematics.

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Chapter 5

Using Graphing Software for Posing Problems in Advanced High School Algebra 5.1 Introduction Many problems of science and engineering require knowledge of the roots of a one-variable polynomial and their location in the (x, y)-plane. For example, one important property of for applications is for its roots to be located to the left of the y-axis or inside the unit disk (e.g., [Leipholz, 1987]). That is, both real and complex roots of P(x) are considered. Different methods exist to decide such properties of without finding the roots explicitly (e.g., [Koenig, 1953; Marden, 1966]). When only real roots are considered, the problem of deciding the number of different (not multiple) roots of P(x) that belong to a given interval was solved by Sturm23 who developed an algorithm (reminiscent of the Euclidean algorithm of finding the greatest common divisor of two integers) of exploring the change of signs of a recursively constructed sequence of auxiliary polynomials called Sturm’s functions (starting from P(x) and its first derivative) at the end points of the interval [Cajori, 1919; Uspensky, 1948]. However, those general methods, while have to be studied by the future extenders of the frontiers of STEM (science, technology, engineering, mathematics) knowledge, are outside the boundary of the mainstream mathematics curriculum, including that of a secondary mathematics teacher preparation program. The goal of this chapter24 is to share a possible use of traditional (and mostly procedural) content of quadratic equations as a background for grade-appropriate mathematical investigations rudimental to real problems of science and engineering. Those rudiments of STEM explorations are pertinent to a secondary mathematics teacher education program. In what follows, a single-variable quadratic equation with two 23 24

Jacques Charles François Sturm (1803–1855) – a French mathematician. This chapter is an extended version of the author’s paper [Abramovich, 2016b]. 105

106 Integrating Computers and Problem Posing in Mathematics Teacher Education

real parameters will be considered as an object of an analytic inquiry aimed at constructing regions in the plane of parameters responsible for a specified location of two real roots about a given interval. In the digital era, explorations of that kind can lead to conceptually rich mathematical activities that are both technology-immune and technology-enabled (TITE). As discussed in Chapter 3, the importance of a TITE problemsolving mathematics curriculum is due to the availability of sophisticated computer programs capable of the variety of symbolic computations and graphic constructions. While the omnipresent use of digital tools in the modern classroom makes procedural skills less demanding, the conceptual component of mathematics teaching and learning may not be neglected. The problem-posing and problem-solving focus of a TITE mathematics curriculum has the potential to maintain the right balance between the procedural and the conceptual. Polynomials of the second degree and the use of their roots in different algebraic explorations are included into the secondary mathematics curriculum. Taking pedagogical advantage of the curriculum, instead of dealing with the two-dimensional plane, one can use the one-dimensional x-axis and study the roots’ location about an interval on the number line. This study may be considered as a prerequisite for real-life STEM investigations. In the classroom setting, even when there exists a formula for explicitly finding the roots in terms of the coefficients of a polynomial, it may be too complex to allow for an effective localization of the roots. This is true even in the simple case of quadratic equations. Although the quadratic formula is not difficult to deal with, determining the location of roots about an interval on the number line using this formula requires solving inequalities involving radical expressions. The use of such inequalities can be avoided due to qualitative methods which allow one not to deal with the roots directly. Those qualitative methods can be taught already at the secondary level under the conceptual umbrella of STEM education, especially when the tools of technology are commonly available. Towards this end, the chapter offers several teaching ideas that include certain rudiments of STEM problems using grade-appropriate mathematical machinery and content. The ideas stem from a capstone secondary mathematics education course that the author has taught to prospective secondary mathematics teachers over the years.

Using Graphing Software for Posing Problems in Advanced High School Algebra 107

An effective approach to the localization of roots of polynomials on the number line is to use what may be called a geometrization of analytic situations through constructing the graphs of the polynomials. It goes back to the 17th century when Descartes25 realized that it is possible to establish one-to-one correspondence between an algebraic equation in two variables and a set of points in the coordinate plane. For example, the equation y = x uniquely describes the straight line which bisects the first and the third coordinate angles in the (x, y)-plane. Indeed, any point on the bisector is equidistant from the coordinate axes and this property of the points can be expressed by the relation x = y – the simplest algebraic equation in two variables. Due to this wondrous insight, algebra and geometry had become connected and the analytic geometry was born. The use of analytic geometry in solving algebraic problems can be demonstrated by using quadratic equations without explicitly expressing their roots through the quadratic formula. Another conceptual alternative to the quadratic formula is to use Vieta’s26 Theorem. These methods, their description in terms of the modern educational constructs, and the extension into problem posing constitute the main focus of this chapter. Briefly, the equivalence of the geometric/graphic method to the construction of Sturm’s functions to be explored in terms of the change of signs at the endpoints of an interval will be demonstrated (section 5.8) in the case of a quadratic equation. 5.2 Location of roots of quadratics about an interval A rather mundane (and perhaps even boring) activity of solving quadratic equations can be significantly enriched by considering same equations but with parameters in place of coefficients. A didactic value of this extension is at least twofold. It enriches secondary mathematics education courses with explorations redolent of real research experience in a STEM field that teacher candidates need to acquire. The Conference Board of the Mathematical Sciences [2012], an umbrella organization consisting of sixteen professional societies in the United States 25

René Descartes (1596-1650) – a French mathematician and philosopher. Franciscus Vieta (1540-1603) – a French mathematician, credited with the introduction of algebraic notation into mathematics. 26

108 Integrating Computers and Problem Posing in Mathematics Teacher Education

concerned, in particular, with the mathematical preparation of schoolteachers, supports this position by noting that those teachers “who have engaged in a research-like experience for a sustained period of time frequently report that it greatly affects what they teach, how they teach, what they deem important, and even their ability to make sense of standard mathematics courses” (p. 65). Also, the extension into equations with parameters makes it possible, by introducing historical perspectives into a mathematics teacher education course, to open entries into a hidden mathematics curriculum comprised of many ideas developed by great mathematical minds of the past. Such ideas can be mapped to the traditional mathematics curriculum enabling mathematical connections to be developed through computer-motivated collateral learning as “the proper use of technology make[s] complex ideas tractable, [and] it can also help one understand subtle mathematical concepts” [ibid, p. 57]. Finally, the blend of digital tools, research-like experience, historical perspectives, and appeal of hidden mathematics curriculum [Abramovich and Brouwer, 2006] allow for a stronger collaboration between mathematicians and mathematics educators towards the dual goal for teacher candidates’ seeing “the larger body of mathematics … arising from the ideas under discussion … [and] appreciating the nature and role of meaning in students’ mathematical learning” [Thompson, Artigue, Töner and de Shalit, 2014, p. 320]. In what follows, various TITE problems associated with the use of the geometric method in algebra (i.e., analytic geometry) supported by computer graphing and symbolic computation software tools will be considered. More specifically, the immediate goal is to determine conditions in terms of the coefficients of the quadratic equation x2 + bx + c = 0

(5.1)

responsible for a specific type of location of its (different) real roots about an interval on the x-axis. The first, technology-immune (TI) part of this problem is to determine how many ways two roots can be located on the number line about an interval. To this end, the concept of combinations with repetitions (e.g., [Vilenkin, 1971; Abramovich, 2017]) has to be considered. For example, when selecting two problems to solve on Saturday out of five given as a weekend homework, the selection of

Using Graphing Software for Posing Problems in Advanced High School Algebra 109

the same problem more than one time does not make sense. However, one’s selection of two donuts out of five types available in a grocery store is seen as creating two-combinations with repetitions out of five objects (thus having the objects repeated in a combination), because allowing same donut type to be selected twice does make sense. Note that in both types of combinations (without and with repetitions), as the above concrete examples illustrate, the order of objects is irrelevant. In general terms, one has to find the number of k-combinations selected out of m different types of objects, allowing a combination to include same object types. In the case of two roots and three intervals into which a given interval divides the number line (Fig. 5.1), the roots have to form two-combinations with repetitions because they may or may not belong to the same interval out of the total of three intervals; that is, k = 2 and m = 3.

Fig. 5.1. Three types of location for real roots of equation (5.1).

The situation can be described in terms of two letters R (the roots; alternatively, objects) and two letters E (the given interval’s endpoints separating three types of roots location) being mutually arranged. There are six such arrangements which are not difficult to be listed: RREE, RERE, REER, ERER, ERRE, and EERR. This is similar to having six outcomes with two heads in the experiment of tossing a coin four times as represented by the number six in the fifth row of Pascal’s triangle (Fig. 1.1, Chapter 1). In the general case of k objects and m types, the number of such arrangements is equal to m = 3 we have

! ! !

! !

!

. In the case k = 2 and

= 6 as the number of permutations of letters in the

word RREE. Note that in finding the above six permutations, technology was not used and in that sense this part of the problem is technology-immune. It

110 Integrating Computers and Problem Posing in Mathematics Teacher Education

prepares one to use a digital graphing tool in constructing all possible cases defined by the six permutations of letters in the string RREE. Technologically, this construction is not automatic and it requires one more pre-technological cognitive engagement to be considered. 5.3 Digital fabrication Digital fabrication as a modern-day educational paradigm has grown out of the need to study a classroom pedagogy of student-computer interaction in order to improve the teaching and learning of STEM disciplines [Gershenfeld, 2005; Walter-Herrmann and Büching, 2013]. This interaction occurs in a cognitive space where abstract mathematical relations and concrete images (described by those relations) meet [Nake, 2013]. Put another way, this is a space where by posing problems the first-order symbols and the second-order symbolism meet enabling problem solving. One can see a connection between the concept of TITE problem and the paradigm of digital fabrication. In order to deal with a mathematical concept as a tool in turning the abstract into the concrete by using technology, one has to consider separately the TI and TE components of a problem in hand. That is, digital fabrication is a TITE problem posing and solving with two distinct parts. A simple example of digital fabrication is the use of two-variable inequalities in constructing an interval “penetrated” by the x-axis. To this end, a system of the inequalities | | , | | can be entered into the input box of the program Graphing Calculator [Avitzur, 2011] capable of graphing images defined by two-variable inequalities. As a result, the program digitally fabricates the (-n, n) interval of thickness 2 (see Figs 5.2–5.7 below where n = 1, 0.05). Now, consider equation (5.1) the coefficients of which satisfy the 4 0 to allow for two real roots, p and q, p > q. For inequality simplicity, a symmetric about the origin interval (-n, n) will be considered. The task is to explore the location of the roots about this interval and partition the plane (c, b) of parameters into the regions corresponding to six different types of the roots’ location about the interval (-n, n). Let , , — a function of variable x with parameters b and c.

Using Graphing Software for Posing Problems in Advanced High School Algebra 111

How can a parabola – the graph of the quadratic function , , – be constructed (alternatively, fabricated) using computer graphing software in some systematic way in order to establish its location about an interval? The graph of a quadratic function with two real roots can be defined as the graph of the relation where, in the context of the Graphing Calculator, the roots p and q can become slider-controlled parameters. To this end, one can define the parameters p = slider(-3, 3) and q = slider(-2, 2) and use them as tools that can be altered in order for the parabola and the interval (-1, 1) to match each of the six permutations of letters in the string RREE. It is in that way that the graphs pictured below in Figs 5.2–5.7 have been constructed. Such graphing activity is a digital fabrication which represents a technology-enabled (TE) part of the problem. It is followed by a technology-immune (TI) component which requires one to be able to interpret each of the six cases analytically. Remark 5.1. When using computer graphing software as a tool of digital fabrication, procedural skills of plotting graphs have to be informed by conceptual understanding of the properties of functions that the graphs represent. The question about ways of digitally fabricating a parabola with specific properties is a second-order question. It requests explanation of what the formal steps that one carries out informally when drawing a graph by hand are. This is similar to how one can explain to a non-native speaker of a language how to form a grammatical construction which a native speaker of the language is able to do at an early age without any formal schooling. In the case of English language, the utterance “I saw him rolling two dice” provides some information without explaining how certain words were modified and put together to communicate an observation. This interplay between the procedural and the conceptual, between informal and formal, is the core of digital fabrication. However, there is a difference between plotting parabolas with different types of intersections of the x-axis and constructing regions, defined by inequalities, in the plane (c, b), where and , corresponding to those different types. In plotting the graph of a

112 Integrating Computers and Problem Posing in Mathematics Teacher Education

quadratic function with a specific property, the visual guides the symbolic. That is, from the image of the location of the graph about the interval, the relations among b, c, and n can be derived. On the contrary, as will be shown in section 5.6, in constructing a region in the plane of parameters, the symbolic enables the visual. That is, the relations among the parameters can be used to create a new type of a diagram from which the behavior of the original parabola can be derived. One can see that we have a conceptual system with feedback where by moving from visual to symbolic to visual, the two visuals can be linked. This shows the real complexity of ideas that comprise technological pedagogical content knowledge [Angeli and Valanides, 2009; Mishra and Koehler, 2006; Niess, 2005] necessary for teaching mathematics as an applied discipline. Such teaching is in the spirit of Montessori [1917] who, as was already mentioned in the book, believed that teachers have to be prepared approaching new concepts by integrating experiment, observation, and formal demonstration. 5.4 Connecting the coordinate plane with the plane of coefficients This section will show how one can explore the above six permutations of letters in the string RREE and, using the results of digital fabrication, express each case in the form of a system of two-variable rational inequalities among b, c, and n. As will be shown in section 5.9, while the geometric/graphic method makes it possible to avoid the use of irrational inequalities, they can be found being hidden among rational inequalities. Moreover, the use of Vieta’s Theorem (formulas) will be shown (section 5.5) as a purely TI alternative to digital fabrication through which another kind of hidden inequalities can be recognized. 5.4.1 The case RREE Both roots of equation (5.1) are smaller than -n. As shown in Fig. 5.2, the inequality , , 0 should hold true and the vertex, x0 = -b/2, 2 of the parabola y = x + bx + c should be located to the left of the line x = -n. Therefore, the following system of inequalities should be satisfied: 0,



.

(5.2)

Using Graphing Software for Posing Problems in Advanced High School Algebra 113

Fig. 5.2. Both roots are smaller than -1.

As a way of asking a second-order question, one may wonder as to why the inequality , , 0 is not included in the description of the case RREE. Noting that the function , , , as shown in Fig. 5.2, strictly increases to the right of the point , , one can conclude that inequality , , extraneous one.

, , , , 0 and, thereby, the 0, if added to inequalities (5.2), would be an

5.4.2 The case RERE One root of equation (5.1) is smaller than -n, another root belongs to , . As shown in Fig. 5.3, the inequalities , , 0 and , , 0 have to be satisfied. Hence n2 – bn + c < 0, n2 + bn + c > 0.

(5.3)

5.4.3 The case REER One root of equation (5.1) is located to the right of the interval , , another root is located to the left of the interval , . As shown in Fig. 5.4, this case implies f(-n, b, c) < 0 and f(n, b, c) < 0. Hence n2 –bn + c < 0, n2 + bn + c < 0.

(5.4)

114 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 5.3. One root is smaller than -1, another root belongs to (-1, 1).

Fig. 5.4. The interval (-1, 1) is inside the “inter-rootal” [Watson and Mason, 2005, p. 40] interval.

5.4.4 The case ERER One root of equation (5.1) is greater than n, another root belongs to the interval , . As shown in Fig. 5.5, this case implies the inequalities f(-n, b, c) > 0 and f(n, b, c) < 0. Hence n2 – bn + c > 0, n2 + bn + c < 0.

(5.5)

Using Graphing Software for Posing Problems in Advanced High School Algebra 115

Fig. 5.5. One root is greater than 1, another root belongs to (-1, 1).

5.4.5 The case EERR Both roots of equation (5.1) are greater than n. As shown in Fig. 5.6, in addition to the inequality f(n, b, c) > 0, the vertex of the parabola y = x2 + bx + c should be located to the right of the line x = n. Hence n2 + bn + c > 0, –b/2 > n.

Fig. 5.6. The interval (-1, 1) is outside the “inter-rootal” interval.

(5.6)

116 Integrating Computers and Problem Posing in Mathematics Teacher Education

Similar to the case RREE, when both roots were also located outside the interval , , the inequality f(-n, b, c) > 0, while true, may not be added to inequalities (5.6). Indeed, the quadratic function f(x, b, c) is strictly decreasing in the interval , and, therefore, f(-n, b, c) > f(n, b, c) > 0. That is, the quadratic inequality in (5.6) implies the inequality f(-n, b, c) > 0. 5.4.6 The case ERRE Finally, both roots of equation (5.1) belong to the interval , . As shown in Fig. 5.7, this case implies f(-n, b, c) > 0, f(n, b, c) > 0, and, to avoid the cases RREE and EERR shown in Fig. 5.2 and Fig. 5.6, respectively, the line of symmetry of the parabola must be crossing the interval , . Hence n2 – bn + c > 0, n2 + bn + c > 0, | b | < 2n.

(5.7)

Fig. 5.7. The “inter-rootal” interval is within (-1, 1).

5.5 Using Vieta’s Theorem Alternatively, as a purely TI component of the activities, inequalities (5.2)–(5.7) can be derived using Vieta’s Theorem for equation (5.1), connecting its roots p and q with the coefficients b and c through the formulas

Using Graphing Software for Posing Problems in Advanced High School Algebra 117

,

.

(5.8)

To this end, three distinct cases of the mutual arrangement of three points on the number line need to be considered. If both p and q are smaller than the number t, then the (obvious) inequalities 0, 0, 2 , are equivalent to the , or 0, inequalities 0, , whence, due to formulas (5.8), 0,

.

(5.9)

If both p and q are greater than the number t, then the (obvious) inequalities 0, 0, 2 , are equivalent to the inequalities 0, , or 0, , whence, due to formulas (5.8), 0,

.

Finally, if , then the (obvious) inequalities 0 are equivalent to the inequalities 0, whence, due to formulas (5.8), 0 .

(5.10) 0, 0, or

(5.11)

Note that the inequality 0 is equivalent to the combination of the inequalities p – t > 0, q – t < 0 and p – t < 0, q – t > 0. It follows from the second pair of inequalities that p < t and q > t, contrary to the assumption p > q. Therefore, the inequality p > q implies only the relations q – t < 0 and p – t > 0, so that, unlike (5.9) and (5.10), inequality (5.11) does not need to be augmented by a condition on b. Now, the above six cases of the roots’ location about the interval , can be revisited using formulas (5.9)–(5.11). 1. The case RREE (section 5.4.1) can be described through inequalities (5.9) by setting t = -n, whence inequalities (5.2).

118 Integrating Computers and Problem Posing in Mathematics Teacher Education

2. The case RERE (section 5.4.2) can be described by setting in (5.9) 0, and (5.11) t = n. This yields the inequalities 2 , 0, which, in comparison with inequalities (5.3) require an additional condition for b. However, inequalities (5.3) imply the inequality 2 and, in that sense, one can say that the last inequality is hidden in inequalities (5.3). Indeed, subtracting the inequality 0 from the inequality 0 yields 2 0, whence (after dividing both sides of the last inequality by the positive factor 2n) the inequalities 0 2 . 3. The case REER (section 5.4.3) can be described by using inequality (5.11) twice: setting t = - n and t = n. This immediately yields inequalities (5.4). 4. The case ERER (section 5.4.4) can be described by setting t = -n in (5.10) and t = n in (5.11). This yields the inequalities 0, 2 , 0. Once again, the inequality 2 is not included, but rather hidden, in inequalities (5.5). Indeed, subtracting the 0 from the inequality 0 yields inequality 2 0, whence (after dividing both sides of the last inequality by the negative factor -2n) the inequalities 0 2 . 5. The case EERR (section 5.4.5) can be described by setting t = n in inequalities (5.10). This immediately yields inequalities (5.6). 6. Finally, the case ERRE (section 5.4.6) can be descried by setting t = n in (5.9) and t = -n in (5.10). This yields the inequalities 0, 2 , 0, 2 which (after noticing that the inequalities b > -2n and b < 2n are equivalent to |b| < 2n) coincide with inequalities (5.7). Remark 5.2. The above-mentioned cases of hidden inequalities were revealed in the context of using formulas (5.9)–(5.11) when one and only one root belongs to the interval , . As shown in Fig. 5.3 and Fig. 5.5, the relationship between the vertex of the corresponding parabola and the right/left end point of the interval, something that a hidden inequality specifies, is automatically embedded into the corresponding visual illustration. This observation underscores the importance of formal reasoning being mediated by an interplay between visual and analytic representations of a mathematical concept. Whereas, generally speaking,

Using Graphing Software for Posing Problems in Advanced High School Algebra 119

the former representation may not be considered rigorous, the latter representation, in the absence of its situational referent in the form of a graph, might bring about symbolic information that has no effect on the outcome of a digital fabrication. By the same token, an analytic interpretation of the behavior of a graph in terms of relations among the three parameters is not straightforward and its outcome depends on the level of mathematical competence of an interpreter. All things considered, it appears that using jointly both representations ensures computational efficiency of the corresponding algorithm which connects the TI and TE components of the problem in question. 5.6 Posing TITE problems in the plane of parameters The next step (a purely TE one) of the problem-posing and problemsolving activities is to partition the plane of parameters (c, b) into the regions corresponding to the above six types of roots’ location about the interval , defined by inequalities (5.2)–(5.7), respectively. Note that the inequality b2 – 4c > 0 should also be taken into consideration when constructing the regions defined by the systems of inequalities (5.2), (5.6), and (5.7) as those systems would still be satisfied if the corresponding parabolas in Fig. 5.2, Fig. 5.6, and Fig. 5.7, respectively, would be located entirely above the x-axis. In the other three cases, as shown in Figs 5.3–5.5, at least one of the inequalities 1 0 (the case n = 1) would imply that the corresponding parabola has points in common with the x-axis. This generalized partitioning diagram (where n is a slider-controlled parameter) is shown in Fig. 5.8. Several TITE problems can be posed using the partitioning diagram as a problem-posing environment. A TE component of those problems may go beyond constructing the diagram to include some additional (digital) fabrication of points and lines. Note that in the context of the Graphing Calculator the point (c0, b0) can be either constructed as the intersection of the lines x = c0 and y = b0, or digitally fabricated through | | ,| for a sufficiently small 0. the inequalities | As will be demonstrated below, once the diagram is constructed, it serves as support system for a TI component of a problem.

120 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 5.8. Regions in the plane (c, b) defined by inequalities (5.2)–(5.7).

Problem 5.1. One can select a point from the (c, b)-plane of parameters, notice to which of the six regions it belongs along with the value of n, use its coordinates as the coefficients of the quadratics, find the roots of the latter (perhaps, using technology), and check to see that their location about the interval , for the chosen n corresponds to the region to which the selected point belongs. This shows the advantage of theory over experiment when selecting parameters of a system with the desired behavior. Problem 5.2. Using the Graphing Calculator, one can check to see that adding to inequalities (5.3) the inequality b > -2n has no effect on the digital fabrication of the region RERE. Likewise, adding to inequalities (5.5) the inequality b < 2n has no effect on the region ERER. This TEbased observation can then be followed by a TI explanation as described in section 5.5. Problem 5.3. The apparent symmetry about the c-axis of the regions RERE and ERER, as well as the regions RREE and EERR can motivate another TI activity. To this end, let the point , ∈ for a certain value of n. This means that equation (5.1) has real roots p and q

Using Graphing Software for Posing Problems in Advanced High School Algebra 121

such that . Multiplying the last chain of inequalities by negative one yields the equivalent chain which shows that the status of the pair , about the interval , is of the ERER type. Furthermore, the equalities 0 and 0 are equivalent to 0 and 0, respectively. In other words, -p and -q, for which are the roots of the equation 0 into which equation (5.1) turns when the point (c, b) is replaced by the point (c, -b). This completes the demonstration of the symmetry of the regions RERE and ERER about the c-axis. Similarly, other visually apparent symmetries about the c-axis in the (c, b)-plane can be demonstrated. Problem 5.4. One can be asked to investigate how the roots behave when the case ERRE turns into that of RREE. The diagram of Fig. 5.8 shows that the two regions converge into the point of tangency of the straight line 0 and the parabola 4 0. Solving the last two equations simultaneously yields , , 2 . At that point, equation (5.1) assumes the form 2 0, whence (i.e., ). That is, the case ERRE bifurcates into the case RREE when both roots simultaneously pass through the left end point of the interval , . Similarly, the case ERRE bifurcates into the case EERR when both roots of equation (5.1) simultaneously pass through the right end point of the interval , . Problem 5.5. One can explore the behavior of the roots of equation (5.1) on the borders of the regions shown in Fig. 5.8. For example, the regions RREE and RERE share the border described by the line where equation (5.1) turns into the equation The roots of the last equation can be found using formulas (5.8): . When p = q we have ; when p > q we have ,

0. .

That is, on the border between the regions RREE and RERE, while one of the roots remains outside the interval , , another root becomes equal to –n. This (perhaps, intuitively clear) conclusion is reflected in the partitioning diagram of Fig. 5.8. Similarly, in order to see what happens when the case ERRE bifurcates into the case REER at the point where the lines

122 Integrating Computers and Problem Posing in Mathematics Teacher Education

0 and 0 intersect, one can find out (both analytically and , 0 where equation graphically) that such point has the coordinates (5.1) turns into 0, whence . Remark 5.3. Note that the points located on the borders of the six regions provide equation (5.1) either with a double root, or, in some cases, with at least one root coinciding with an end point of the corresponding interval. Problem 5.6. By changing n, one can observe in the (c, b)-plane that as n > 0 decreases, the region ERRE shrinks. Requesting an explanation of this phenomenon leads to another TITE problem-solving activity. To this end, one can be asked to describe analytically the region ERRE in terms of inequalities among b, c, and n. Such a description can be done as a combination of analytic reasoning and symbolic computations using, once again, Wolfram Alpha. In order to find the smallest value of the crange for that region, one has to solve simultaneously the equations 0 and 0 to get . The largest value of the c-range results from solving another pair of equations, 4 , to get . That is, in the region ERRE we have 0 and . When 0 we have ; when 0

we have 2√

. This analytic description

suggests that the region ERRE shrinks as n decreases, and it is attracted by the origin when → 0. Remark 5.4. A second-order question that may be asked about the partitioning diagram of Fig. 5.8 is to request the following explanation: why is it important to know how to construct regions in the plane of parameters responsible for a particular type of behavior of a system described by an equation the solution of which depends on the parameters? To answer this question note that in the context of STEM education, when exploring mathematical models, a student must have experience with phenomena that go beyond pure intuition [American Association for the Advancement of Science, 1993]. For example, it may not be intuitively clear that with the decrease of the interval’s length, the

Using Graphing Software for Posing Problems in Advanced High School Algebra 123

region ERRE (when the interval contains both roots of a quadratic equation) shrinks and becomes attracted by the origin in the plane of parameters. Nowadays, the exploration of non-intuitive situations can be greatly enhanced by the use of technology. At the same time, their formal demonstration requires analytic reasoning skills which could be enhanced but not supplanted by the use of symbolic computations. Pedagogically speaking, this provides an opportunity to pose a variety of TITE problems in a mathematics classroom, including that of prospective secondary teachers. Having a mechanism for the appropriate selection of parameters of a mathematical model that provide the required properties of the behavior of the model, demonstrates the value of TITE problemposing and problem-solving activities in advancing STEM education at different levels. 5.7 Geometric probabilities and the partitioning diagram The use of the partitioning diagram of Fig. 5.8 can be extended to include problems dealing with geometric probabilities. In order to explain the concept of geometric probability (that goes back to Buffon’s needle problem briefly mentioned in Chapter 1, section 1.6), consider a trial that consists in random choice of a point over the region Ω. The task is to find the probability that this point belongs to the region ⊆ Ω. Because the number of outcomes is uncountable, one cannot define (theoretical) probability as the ratio of the favorable outcomes to the total number of outcomes. In order to define geometric probability, let us assume that the outcomes of the trial are distributed uniformly. This means that if the region Ω is decomposed into a finite number of equal parts , and Τ is the event that a random point falls into the region , then the outcomes Τ are equally likely and it is in that sense that the probability distribution over Ω is uniform. In other words, the probability of a point falling in any part of the region Ω is proportional to the measure of and does not depend on its location and shape. Therefore, by definition, the probability that a point thrown randomly onto Ω will fall in is equal to the ratio , where and Ω are the measures of the corresponding regions expressed through the corresponding units of measurement. This

124 Integrating Computers and Problem Posing in Mathematics Teacher Education

definition allows one to pose a variety of problems the solution of which requires finding areas of different two-dimensional shapes either by a geometric formula or by integration. As an example, consider Problem 5.7. The coefficients of equation (5.1) are chosen at random from the rectangle , :| | 2 ,| | . Find the probability that: a) both roots belong to the interval , ; b) one root belongs to the interval , , another root is smaller than ; c) one root belongs to the interval , , another root is greater than ; d) equation (5.1) does not have real roots. Considering this problem as a TITE one, its TI part consists of developing an algorithm for constructing the image of the rectangle from which the points (c, b) are randomly selected to serve as coefficients of equation (5.1). First, as one can see from Fig. 5.9 (where such rectangle has been constructed), the rectangle has relatively thick sides having a different appearance than what the graphing program does when lines and curves are defined by a two-variable equation. As before, the notation (x, y) will be used in place of (c, b). In order to control the thickness of the vertical line x = n2, one has to introduce parameter 0 2 so that the distance between the coordinate x and the number n is | smaller than . The last condition is defined by the inequality | . For example, when 0.1 the program produces a vertical strip of thickness 0.2, not bounded from either above or below. But the ycoordinates of the points of the strip have to be smaller than 2n and | greater than –2n. Thus, the system of the inequalities | 0.1, | | 2 defines a vertical strip of height 4n and thickness 0.2. The opposite side of the rectangle passes through the point x = –n2. Therefore, in order to have both n2 and –n2 in a single inequality, the coordinate x we have or has to be replaced by | | because when | | . That is, the system of the inequalities || | | 0.1, | | 2 defines two vertical sides of the rectangle. Likewise, the system of the define the remaining sides of the inequalities || | 2 | 0.1, | | rectangle with length 2 and thickness 0.2.

Using Graphing Software for Posing Problems in Advanced High School Algebra 125

Fig. 5.9. Augmenting the partitioning diagram with a rectangle.

Augmenting partitioning diagram of Fig. 5.8 by rectangle (as shown in Fig. 5.9) is the TE part of the problem as through the use of computer graphing software one can clearly see the geometric relationship between the regions ERRE, RERE, ERER, and the rectangle. The area of the rectangle with the side lengths 2n2 and 4n is equal to 8n3. The area or the region ERRE can be found as the difference between the area of the rectangle and the joint area of the parabolic segments and the regions RERE and ERER which are the part of the rectangle. This difference is 4

. Therefore, the

probability sought in part a) is equal to

; the probability sought

equal to 8

4

2

2√

; the probability sought in part c) is equal to

in part b) is equal to

; the probability sought in part d) is equal to note that

. Finally,

1.

5.8 Making mathematical connections As was mentioned in section 5.1, a classic method of deciding the number of different roots that belong to a given interval is due to Sturm and the following description of the method is borrowed from [Cajori, 1919]. Let f(x) = 0 be an equation which does not have equal roots in the and interval (a, b). Let us divide f(x) by its first derivative, find the negative of the remainder, . This division process continues (changing the sign of each new remainder before using

126 Integrating Computers and Problem Posing in Mathematics Teacher Education

it as a divisor) until a remainder not depending on x is reached and its 0. This recursive sign is changed as well to have process is reminiscent of the Euclidean algorithm of finding the greatest common divisor of two integers when the division process, through which the sequence of remainders is calculated, stops once the zero remainder is reached. The functions , , ,…, are called Sturm’s functions. Sturm’s Theorem states that the difference between the number of sign changes in the sequence of Sturm’s functions when x = a and x = b is equal to the number of real roots of the equation f(x) = 0 in the interval (a, b). 5 6 0 and the As an example, consider the equation interval (0, 5). Sturm’s functions in this case are 5 as 5 6 2 5 6, 2 5, and . When x = 0 the sequence

0

6,

0

5,

0

has two

sign changes (from + to – and then back to +); when x = 5 the sequence 5 6, 5 does not change sign. According to 5, 4 5 6 0 has 2 – 0 = 2 real roots Sturm’s Theorem, the equation in the interval (0, 5). In the general case of quadratic equation (5.1), assuming 4 0, one can recognize the equivalence of the geometric/graphic and Sturm’s methods, both free from the explicit use of the quadratic formula (or Vieta’s formulas). In that case, the sequence consists of three , its derivative 2x + b, polynomials: the quadratic trinomial and the negative of the remainder obtained by dividing by 4 . Indeed, one can check to see that within the 2x + b; that is, relation among the dividend, the divisor, the quotient, and the remainder: 2 4 , the negative of the remainder turns out to be the 1/4 multiple of the discriminant, 4 , of the quadratic trinomial. As an aside, note that the (apparently less commonly used) identity 2 4 not only connects a quadratic trinomial, its first derivative 2 , and the discriminant 4 , but it also shows that the positive discriminant does provide the trinomial with two different real roots. Indeed, the

Using Graphing Software for Posing Problems in Advanced High School Algebra 127

equation 2 4 has a double root when 4 4 < 0, then it follows from the above identity 0. However, when that > 0 for all real values of x so that the equation 0 does not have real roots. Alternatively, one can enter into the input box of Wolfram Alpha the command “solve x^2+bx+c=(2x+b)(px+q)+r in p, q, r” to get the solution p = 1/2, q = b/4, and r = (1/4)(4c – b2). Note that the inequality 4 0, by making negative r positive, does guarantee the existence of two real roots of the equation 0. According to Sturm’s Theorem, the number of real roots in the interval , is equal to the difference between the number of sign changes of the three polynomials at the points x = -n and x = n. Due to the discriminant inequality (which does not depend on x), only the first two polynomials may be subject to sign changes. Consider the case ERRE (section 5.4.6), when such difference has to be equal to the number 2. Because 4 0, for x = -n we should 0 (the inequality of the opposite have the inequalities sign would not be consistent with two sign changes allowing either or combinations) and -2n + b < 0 (in other words, the negative value of the first derivative implies that the quadratic function decreases at the left end point of the interval); when x = n, no change of sign is required implying the inequalities 0 (the positive value of the quadratic function at the right end point of the interval implies exactly two intersections of the corresponding parabola with the interval) and 2n + b > 0 (the positive value of the first derivative at the point x = n confirms the increase of the quadratic function). That is, the case ERRE, according to Sturm’s Theorem, requires that inequalities (5.7), augmented by the inequality 4 0, hold true. This shows the equivalence of the geometric/graphic and Sturm’s methods, each of which does not require either the explicit or implicit knowledge of the roots. Furthermore, interpreting the behavior of Sturm’s functions in terms of the behavior of a quadratic function enhances the TI component of the activity. Similarly, other cases of the roots’ location about the interval , can be used as problem-posing activities based on Sturm’s method.

128 Integrating Computers and Problem Posing in Mathematics Teacher Education

Remark 5.5. Technological availability of symbolic computations makes it possible, without much difficulty, to develop Sturm’s functions . Just as in the proceeding from the cubic polynomial above case of the quadratic trinomial, using Wolfram Alpha one can first find the remainder 9 when dividing the 3 cubic polynomial by its first derivative, and then divide the derivative, 3 2 , by to get the negation of the second remainder in the form 9 4 4 18 27 . 4 3 Arriving at the remainder which does not depend on x terminates the recursive process of division. The expression 4 4 18 27 is the discriminant of the cubic polynomial (information provided, among other sources, by Wolfram Alpha), which is equal to zero in the case of three equal roots (e.g., for the equation 3 3 1 0). Conditioning the discriminant to be positive enables two things: 1) the cubic polynomial has three real roots; 2) the inequality 0 holds true whatever the interval on the x-axis as does not depend on x. 5.9 Revealing hidden concepts through collateral learning The use of graphs in defining the six cases of the roots location about an interval through rational inequalities (5.2)–(5.7) or formulas (5.9)–(5.11) made it possible to avoid solving inequalities involving radical expressions. In the case of quadratic equations, with an easy algorithm of finding roots (i.e., the quadratic formula), the computational power of a digital tool does not differentiate between dealing with radical or rational relations. Yet, an unwarranted use of this power might lead to significant difficulties (including just the typing errors) already in the case of polynomial equations of the second order. Furthermore, fostering the culture of computational efficiency of a mathematical investigation is an important skill that has to be emphasized as appropriate through the teaching of STEM-related disciplines. The issue of reducing computing complexity becomes meaningful in the context of STEM education if a computational environment used offers an improvement over the traditional one in terms of how calculations could be performed.

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By using an efficient computational algorithm, one can gain better conceptual understanding of mathematical relationships that underpin the algorithm. This also gives new meaning to the concepts involved. In particular, in the case of a quadratic equation, by comparing two computational algorithms, one based on irrational inequalities and another based on rational inequalities (whatever the method), the notion of hidden inequalities can be discussed. An interesting aspect of using digitally fabricated graphs as well as formulas (5.9)–(5.11) versus the use of the quadratic formula is that one type of inequalities may be found being hidden within another type (see also [Abramovich and Connell, 2015]). Similar examples of one procedure/concept being hidden within another procedure/concept are the repeated addition being hidden within the multiplication, repeated subtraction being hidden within the division, or Sturm’s Theorem in the case of a quadratic equation being hidden behind the geometric/graphic method. That is, a concept which provides a less cumbersome (or a less cognitively demanding) computational procedure hides a more complicated (or a more general) procedure within an efficient (or easy to use) one. To clarify (in addition to two illustrations of section 5.5 where the notion of hidden inequalities was discussed), consider the case RREE (Fig. 5.2) as an example. The recourse to the roots of equation (5.1) leads to the inequality

–

√

√

whence 4

2 .

(5.12)

Because 4 0 implies √ 4 > 0, inequality (5.12) is equivalent to the following system of rational inequalities 4 0, 2 0, 4 2 . Simple transformations of the last two inequalities yield inequalities (5.2) 4 0 (enabling two real roots) was for which the condition assumed. It is in that sense that one can say that irrational inequality (5.12) is hidden in rational inequalities (5.2). Similar problems concerning the relations of equivalence between irrational and rational inequalities can be posed in other five cases of the roots of equation (5.1) location about the interval (-n, n).

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In such a way, under the umbrella of the educational construct of collateral learning [Dewey, 1938], revealing hidden inequalities (or any hidden procedure/concept for that matter, like Sturm’s Theorem) serves the purpose of educational efficiency “not only in reaching the projected end of the activity [e.g., digital fabrication] immediately at hand, but even more in securing from the activity the learning which it potentially contains” [Kilpatrick, 1918, p. 334]. This also brings to mind the notion of the didactical phenomenology of mathematics as “a way to show the teacher the places where the learner might step into the learning process of mankind” [Freudenthal, 1983, p. ix]. The notion of collateral learning encourages teachers to make connections among seemingly disconnected ideas by revealing to students hidden mathematics curriculum – a didactic approach to the teaching of mathematics that motivates learning in a larger context and rejects “the greatest of all pedagogical fallacies … that a person learns only the particular thing he is studying at the time” [Dewey, 1938, p. 49]. The recurrent procedure, reminiscent of the Euclidean algorithm of finding the greatest common divisor of two integers, of constructing the sequence of Sturm’s functions designed to decide the number of roots within a given interval does belong to a hidden mathematics curriculum of secondary teacher education. 5.10 Summary This chapter has demonstrated how solving quadratic equations, a rather routine module of secondary mathematics curriculum, can be conceptually enriched by providing learners with TITE problems rudimental to true explorations encountered in science and engineering research. Such enrichment comes from considering coefficients of the equations as parameters, something that enables a dynamic perspective on essentially static entity of school mathematics. The goal of explorations was to define conditions in terms of the parameters enabling a specific location of real roots of a quadratic equation about a given interval. To this end, a geometric approach to algebra supported by computer graphing and symbolic computation software tools was considered. This allowed for the construction of a diagram in which the plane of parameters was partitioned into six regions each of which corresponds to one of the six types of the roots’ location about an

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interval. Using the diagram, many TITE problems can be posed. In particular, when a pair of coefficients of a quadratic equation is randomly selected, one can compute the likelihood of this pair’s belonging to a specific region, thereby, determining the geometric probability of a desired outcome. The notion of hidden inequalities and the relationship between the proposed geometric approach and the classic method by Jacques Charles François Sturm were discussed in the collateral learning fashion. The next chapter deals with the relationship between problem posing and Einstellung effect – a state of mind when one is tempted to use a workable problem-solving strategy in a new situation when the strategy is either ineffective or not applicable to the situation at all.

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Chapter 6

Einstellung Effect and Problem Posing 6.1 Examples of Einstellung effect This chapter deals with the relationship between problem posing and the concept of Einstellung effect. The latter is a psychological phenomenon evinced by an individual through a tendency (mindset) to use previously learned workable problem-solving strategy in situations that either can be resolved more efficiently or to which the strategy is not applicable at all. For example, if one learns to carry out simple multiplication of two onedigit numbers by reducing it to repeated addition of the second factor so that 6 2 2 2 2 2 2 2 12, 2 3 3 3 6, 3 4 4 4 4 12, 3 5 5 5 5 15 (knowing how to count by twos, threes, fours, and fives), the case 5 7 is likely to be treated in the same way, that is, by adding 7 five times. Instead, one can still count by fives (which is easier than counting by sevens) if noting that 5 7 7 5 5 5 5 5 5 5 5 35 and thus can avoid counting by sevens. However, by not being able to avoid counting by sevens, one cannot bypass the mindset developed through the automatism of the previous counting practice and, therefore, has to use more complicated repeated addition. Another example of Einstellung effect deals with unjustified use of proportional reasoning after an automatic problem-solving practice when it was necessary to find the number of boxes for 36 donuts knowing that two boxes are needed for 24 donuts. In that case, finding x from the proportion 24:36 = 2:x yields x = 3. Likewise, 96 donuts require eight boxes because 24:96 = 2:x, whence x = 8. That is, the ratio of donuts to boxes is 12 to 1. Similar is the case of finding the number of rods needed to build a 25-step ladder knowing that the number of rods used for the 3step ladder is nine (Fig. 6.1). Once again, one can create a proportion 3:9 = 25:x whence x = 75 – the number of rods in a 25-step ladder. One can then try to apply proportional reasoning to the ladder with a rod at its bottom (Fig. 6.2) noting that this time, the number of steps is 133

134 Integrating Computers and Problem Posing in Mathematics Teacher Education

four and the number of rods is ten. So, if one has to find the number of rods for the 25-step ladder with a step at the bottom, the proportion 4:10 = 25:x yields x = 125/2 – a non-whole number. At the same time, had we asked for the number of rods needed for a 20-step ladder, the answer would be 50 rods and the incorrect solution, based on the use of proportional reasoning, for which the case with 25 steps serves as a counter-example, could be easily overlooked.

Fig. 6.1. A ladder with no step at the bottom.

Fig. 6.2. A ladder with a step at the bottom.

Einstellung Effect and Problem Posing

135

Whereas a counter-example is a helpful tool in demonstrating one’s erroneous thinking, it does not explain the conceptual source of error. In other words, while a counter-example can show the occurrence of Einstellung effect, it does not explain how to avoid an error in similar situations. How can one show that the elements of two sets of numbers are indeed in the same ratio? One way is to show that the numbers form a relationship when each element of one set is a constant multiple of the corresponding element of another set. That is, the variables x and y are in the same ratio if there exists number n such that y = nx. For example, if a box includes 12 donuts, then the number of donuts (d) and the number of boxes (b) form the relationship d = 12b. Likewise, because each step in the ladder pictured in Fig. 6.1 requires three rods, the number of rods (r) and the number steps (s) are in the relation r = 3s. At the same time, because of the bottom step (rod) in the ladder pictured in Fig. 6.2, the relation between r and s for that ladder is r = 3s + 1. Therefore, whereas in the case of the former type of ladder proportional reasoning can be used, this reasoning when applied to the latter type of ladder yields a wrong result, regardless whether the result is a whole number or not.

Fig. 6.3. Counting the number of rods in the 5x5 grid.

A more complicated case is shown in Fig. 6.3, where the task is to count the number of rods that form the 5 5 grid. Without advanced

136 Integrating Computers and Problem Posing in Mathematics Teacher Education

conceptual understanding of the counting algorithm, one can count 12 rods within the 2 2 grid and then assume that proportional reasoning (of which one has only basic conceptual understanding) may be applied to the case of counting rods within any larger square grid. From this assumption, an incorrect solution follows; that is, 4:12 =25:x, whence x = 75. Knowing that the same number was found through counting rods in a 25-step ladder, one can even wonder whether there is a connection between counting rods in the 25-step ladder and in the 5x5 grid and be tempted to make a connection thus developing misconception further. A counter-example can show that the result is incorrect. Indeed, when 5 is replaced by 1 (a single square) the proportional reasoning yields x = 3, something that is obviously untrue. Nonetheless, through a failed connection a counting algorithm could be developed. One counting algorithm (which is not unique) may stem from an observation that the grid in Fig. 6.3 comprises five combs with six teeth being supported by the base with five rods plus another five rods closing the far-right comb. Because each such comb consists of 2 ∙ 5 1 11 rods, the 5 5 grid comprises 55 + 5 = 60 rods. A practice in finding an alternative counting algorithm within this or other problems can help one to avoid Einstellung effect. Another counting algorithm can follow from the failed connection: as can be seen in Fig. 6.3, there are five 5-step ladders four of which share five rods with the adjacent one, and missing one of the five bottom rods of the grid; therefore, the total number of rods within the 5x5 grid can be found as follows: 5 ∙ 15 4 ∙ 5 5 60. This can lead to posing new problems on finding alternative counting algorithms. For example, knowing how to count correctly the number of rods in the ladder with the step at the bottom through the formula r = 3s + 1, the case s = 5 yields r = 16 from where the algorithm follows: 5 ∙ 16 4 ∙ 5 60. This is another source of posing new problems: turning an incorrect answer into a thinking device that leads to new problems. For example, one can use a spreadsheet to generate numeric sequences representing both incorrect (using proportional reasoning) and correct number of rods needed for the square grids of different size (see Fig. 6.4) and use these sequences to pose the following problems.

Einstellung Effect and Problem Posing

137

Problem 6.1.1. What is the smallest grid for which the error in the number of rods found through proportional reasoning exceeds 100? (To solve this problem, one has to find the value of 3n2 – 2n(n + 1) – the difference between the incorrect and correct number of rods within the square grid of size n, and then to find the smallest n for which this difference, n2 – 2n, exceeds 100). Problem 6.1.2. Is it possible for an error to exceed 150%? (The answer to this question is in the negative: one has to show that the ratio tends to

/



being smaller than



(=150%); indeed,

).

Problem 6.1.3. What is the size of the grid for which the error is 15 rods? (To answer this question, one has to solve the equation n2 – 2n = 15 whence n = 5). Problem 6.1.4. What is special about the number sequence n2 – 2n that represents the error? Find as many properties of this sequence as you can. (One property of the sequence n2 – 2n is that its terms, like 15, are 1 2 one smaller than a square number, as 1 2 . In particular, unlike the sequence 1 1, the terms of which may include prime numbers being one greater than a square number (like 17 or 37), the former sequence consists of composite numbers only. Problem 6.1.5. Consider any three consecutive integers. Why does the product of the first and the third integers always represent a square number diminished by one? (To answer this question, one has to show that such integers have the form n, n – 1, n – 2 and the product in question is n2 – 2n. Therefore, in order for the quadratic equation n2 – 2n = k to have a positive integer solution, its discriminant 1 + k has to be a perfect square, m2, whence n(n – 2) = k = m2 – 1).

138 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 6.4. A spreadsheet for posing problems on comparing two solutions.

The above five problems serve as an example of how negative affordances of Einstellung effect can be used to provide conceptual queries (alternatively, the second-order questions) into the sources of errors from where collateral learning opportunities can be developed. Some facts, like the existence of at least two composite numbers among three integers one of which is a perfect square may look trivial. Yet, such facts can be better understood when making mathematical connections through posing problems motivated by advanced conceptual understanding of errors that underscore Einstellung effect. 6.2 Water jar experiments and Einstellung effect In the United States, the study of Einstellung effect was first carried out by Luchins [1942] in the context of water jar experiments. This study was motivated by his teacher Max Wertheimer – one of the founders of Gestalt theory [Ellis, 1938] with its focus on the study of insight problems [Dunker, 1945]. The solution of an insight problem often requires the ability to bypass the mindset developed through practicing a specific problem-solving strategy. Water jar experiments involved three non-graduated jars of given capacity and the task for a subject was to obtain a required volume of liquid by filling the jars with water and pouring it from one jar to another until the task is completed. In the context of paper-and-pencil water jar arithmetic problems, it was found

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that after subjects have practiced several times an identical solution strategy which may not necessarily be the most effective in all cases (or even not applicable to some cases), they develop a mindset that prevents seeing a simpler solution when such solution exists. For example, having the jars of capacity 36, 11, and 3 units of water, one can obtain 16 units through the following subtraction 36 – 11 – 3 – 3 – 3 = 16 which appears to be a workable strategy. But in the case of obtaining the same 16 units from the jars of capacity 42, 11, and 5 units, whereas the same strategy, 42 – 11 – 5 – 5 – 5 = 16, works, there is a way to obtain 16 units through a simple addition as 11 + 5 = 16. At the same time, having the jars of capacity 42, 10, and 5 units makes this (or any other) strategy being not applicable to the target of 16 units. It is one’s inability to see a simpler solution because of the earlier training in and success with a specific problem-solving strategy or failure to recognize limitations of the strategy that lead to Einstellung effect. In the words of Luchins [1942], “Einstellung – habituation – creates a mechanized state of mind, a blind attitude toward problems; one does not look at the problem on its own merits but is led by a mechanical application of a used method” (p. 15, italics in the original). The water jar Einstellung test was originally administered by Zener and Duncker in Germany in the 1920’s [Levitt, 1956]. However, already at the end of the 19th century similar experiments took place at the University of Berlin as Max Wertheimer, in the context of water jar tests, referred to experiments (probably using weights) by Müller and Schumann [1898] that he encountered there as a student [Luchins and Luchins, 1970]. Over the years, different variations of water jar experiments were administered by psychologists [Miller, 1957; ValléeTourangeau, Euden and Hearn, 2011] towards the development of learning environments conducive to the reduction of the dependency of problem solvers on a specific successful experience. In particular, Vallée-Tourangeau, Euden and Hearn [2011] found that when a subject, instead of using arithmetic (as shown above), participates in real experiments using jars and water (with a facet as its source), the effect of Einstellung can be reduced. While it appears that such finding reflects on subjects with low arithmetical skills, it is not surprising because, as was mentioned earlier in the book with reference to Vygotsky [1978], using

140 Integrating Computers and Problem Posing in Mathematics Teacher Education

“the first-order symbols … [facilitates acquiring] the second-order symbolism” (p. 115), which, in the case of mathematics, involves an abstract notation system and grade-appropriate operations within the system. This shows an interesting interplay between the physical and the symbolic. In fact, in many instances, mathematics was motivated by the need to explain things observed at the physical level. In one such instance (as discussed in Chapter 1, section 1.3), the theory of probability evolved from the games of chance (that were very popular in the 17th century) when gamblers’ long-term observations seemed to be counterintuitive and called for explanation based on a disciplined inquiry into the nature of the games. To this end, using algebra, one can write the equation 42 – 10x – 5y = 16 which is equivalent to the equation 10x + 5y = 26 and see that the latter does not have a solution in whole numbers x and y because divisibility by 5 holds true for its left-hand side only. Put another way, when information sought is difficult (or impossible) to find (e.g., obtaining 16 units of water from the jars of capacity 42, 10, and 5 units), the explanation of this difficulty might be the only way to respond to request for information. A classic example (already discussed in Chapter 1, section 1.4) can be drawn from number theory. When it was difficult (in fact, was not possible) to find even a single example of extending the solvability of the equation z2 = x2 + y2 in integers to higher powers, Fermat stated (without presenting proof, yet claiming he had one) that it is impossible to represent the n-th power of a natural number as a sum of two like powers, when n > 2. After some 150 years with no proof available, Euler, in an attempt to make a significant step towards proof, formulated an alternative conjecture: for n > 2 it is not possible to represent the nth power of a natural number through the sum of fewer that n like powers. Euler’s conjecture, which, if true, would imply that Fermat was correct, could have resulted from belief that the (true) equality 63 = 33 + 43 + 53, being computationally quite simple, cannot be replicated for fewer than three cubic addends. However, unlike Fermat’s conjecture, counter-examples to Euler’s conjecture (shown in Chapter 1, section 1.4) were eventually found via computer search. Once again, one can see that a counter-example (which may be purely computational) is helpful only for negating a statement; however, in order to analytically prove a statement, conceptual explanation is required. Only at the end of

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the 20th century, a conceptual explanation of why it is indeed impossible to represent the n-th power of a natural number as a sum of two like powers when n > 2 was found [Wiles, 1995]. To conclude this section, note how differently one can describe two famous conjectures in terms of Einstellung effect and problem posing. Euler’s conjecture is an example of Einstellung effect in posing a problem when experience with unworkable cases created a belief (mindset) which turned out to be mistaken as the problem statement was eventually negated. Fermat’s Last Theorem is an example of posing a problem as a way of avoiding a possible Einstellung effect (rooted in the positive experience with the Pythagorean theorem) when the problem statement was eventually confirmed. 6.3 Posing and solving problems as a remediation of Einstellung effect In mathematics education, the effect of Einstellung on problem-solving performance of students at all grade levels has been acknowledged in various publications [Pólya, 1962; Radatz, 1979; Matz, 1982; Tsamir and Bassini, 2004; Abramovich and Ehrlich, 2007]. Most notably, this acknowledgement can be found in [Pólya, 1962]: “Human nature prompts us to repeat a procedure that has succeeded before in a similar situation” (p. 63). Such a negative affordance of successful procedural experience, which, otherwise, is a positive quality, calls for a pedagogy that encourages conceptual understanding and cognitive flexibility. In its then groundbreaking document, the National Council of Teachers of Mathematics [1989] described the need for change in approaches to mathematics instruction by suggesting that teachers should help their students to “foster mathematical insight” (p. 15) through solving problems, including those self-posed. The word insight implies the use of productive thinking [Wertheimer, 1959] as opposed to reproductive thinking responsible for Einstellung effect. Because mathematics instruction considers important for students to use prior knowledge in order to build new knowledge, the negative affordance of prior knowledge (alternatively, expertise) should be taken into account when working with teacher candidates.

142 Integrating Computers and Problem Posing in Mathematics Teacher Education

The flexibility of thought is also an important condition for creativity and insight without which mathematical problem solving remains a challenge for way too many students. Indeed, Luchins [1942] noted that in the schools focusing on drill and practice students “were not accustomed to being taught one method and expected to seek for, or use other methods” (p. 90). However, already in the mid-20th century it was also observed that the schools using progressive teaching methods assist learners in being more creative and less dependent on the rigidity of thinking [Miller, 1957]. Nonetheless, the perspective on mathematical problem solving grounded in the ‘single question (by teacher) – single answer (by student)’ tradition of teacher-student interaction can still be observed in a mathematics classroom of the 21st century. That is why, nowadays, future teachers “learn to ask good mathematical questions … and to look at problems from multiple points of view” [Conference Board of the Mathematical Sciences, 2001, p. 8] towards enabling students to “check their answers to problems using a different method … and identify correspondences between different approaches” [Common Core State Standards, 2010, p. 6], as well as “to analyze problem situations, select appropriate strategies, and reason quantitatively” [Association of Mathematics Teacher Educators, 2017, p. 106]. Furthermore, good questions by teachers can turn classroom dynamics around and seek such questions from students, thus defying the hidden curriculum of the traditional classroom in which students’ role is to answer teachers’ questions, rather than to ask their own questions [Tizard, Hughes, Carmichael and Pinkerton, 1983; Slater, 2003]. Research on the nature of mathematical thinking and problemsolving abilities indicates that “having a good mathematical mind involves the ability to think in a flexible way” [Dreyfus and Eisenberg, 1996, p. 280]. Through teaching that develops such ability at the precollege level, “it is possible to change students’ views from a fixed mindset [leading to Einstellung effect] to a growth mind-set in ways that encourage them to persevere in learning mathematics” [Conference Board of the Mathematical Sciences, 2012, p. 9]. Likewise, the modernday signature pedagogy of undergraduate mathematics includes the notion that solving a problem through “different approaches … challenges students’ misconception that there is just one technique or one

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solution to a mathematical problem” [Ernie, LeDocq, Serros and Tong, 2009, p. 265]. One teaching method with potential to eliminate and clarify misconceptions is associated with problem posing, “an activity [of asking good questions either oneself or others] that at the heart of doing mathematics” [National Council of Teachers of Mathematics, 1989, p. 138]. As mentioned in the first two chapters of this book, problem posing, in general, goes back to the 15th century Italy and can also be recognized as one of the pillars of Montessori’s [1965] approach to teaching encouraging children to pose their own problems and, in doing so, to “advance upon this road of independence” (p. 28). Proceeding from the viewpoint that problem solving and problem posing are two sides of the same coin [Singer, Ellerton and Cai, 2015] one can argue that mathematics pedagogy which integrates the two activities, by bridging procedural and conceptual knowledge (Chapter 4) has the potential to reduce the Einstellung effect typically associated with procedural competence. Below are examples drawn from the author’s work with mathematics teacher candidates. 6.4 Posing problems for water jar experiments using a spreadsheet Fig. 6.5 shows the spreadsheet confirming that pouring water from the full 30-unit jar (cell A1) three times (cell E5) into the 4-unit jar (cell G1) and one time (cell E3) into the 7-unit jar (cell E1) leaves 11 units of water (cell F1) in the 30-unit jar. At the same time, the sum of the numbers in cells E1 and G1 equals to the number in cell F1. The numbers, 4 and 2, in rows 4 and 6 are irrelevant because a customary (not insightful) solution should always be associated with row 5 where the unity in cell C5 indicates that 11 units of water is required to obtain only one time. The spreadsheet shown in Fig. 6.6 generated three customary solutions in the case when 11 units of water has to be obtained using the jars of capacity 40, 5, and 2 units. The three solutions are located in row 5 showing the following relations: 11 = 40 – 24 – 5, 11 = 40 – 14 – 15, and 11 = 40 – 4 – 25. Yet, an insightful solution is 11 5 3 ∙ 2, a linear combination of the numbers in cells E1 and G1, and it has been hidden behind the three customary solutions as another manifestation of Einstellung effect.

144 Integrating Computers and Problem Posing in Mathematics Teacher Education

The programming details for the spreadsheets of Figs 6.5 and 6.6 are presented in Appendix.

Fig. 6.5. Showing 11

30

3∙4

1 ∙ 7 but hiding 11

4

7.

Fig. 6.6. Showing three customary solutions but hiding 11

5

3 ∙ 2.

6.5 Einstellung effect in finding areas on a geoboard One of the key principles of Gestalt psychology, which regards Einstellung effect as an obstacle to productive thinking, is reification.

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This principle deals with the constructive aspect of perception by which one can recognize in an image more concrete information than the image, in the absence of one’s insight, conveys. Geometry presents a perfect case for the application of the reification principle, in particular, through geoboard activities. Geoboard is a learning environment introduced by Gattegno [1971] for exploring basic geometric ideas at the primary level. On a geoboard, a linear unit is a side of a unit square the vertices of which are the four pegs closest to each other. As an example, consider quadrilateral ABCD (Fig. 6.7) the area of which is to be found. The reification principle suggests that one can see quadrilateral ABCD as embedded into a rectangle comprised of eight unit squares, thus having area measured by eight square units. This enables one to find area of the quadrilateral in an indirect way by subtracting from area of the rectangle the areas of three triangles that do not belong to the quadrilateral but do belong to the rectangle. In that case, the areas of the three triangles are halves of the corresponding rectangles for which a side of the quadrilateral is a diagonal. Thus, the area of quadrilateral ABCD can be found as follows: 8  ( 1 1  1  3  1  2  1  2)  4 . 2 2 2 2 When this problem-solving strategy is shown to elementary teacher candidates, the emphasis on taking half when computing areas of triangles that do not belong to the original shape often results in Einstellung effect even when teacher candidates use an alternative strategy without seeing quadrilateral ABCD as embedded into a rectangle. Rather, the same reification principle suggests that the quadrilateral consists of eight parts labeled a, a1, b, b1, c, c1, d, d1 in Fig. 6.7. However, perhaps due to the lack of conceptual understanding of one-half and seeing it as just one of the two pieces into which the whole is partitioned, some teacher candidates label each of the eight parts as 1/2 and, nonetheless, arrive to the correct answer, 4. When asked to explain their answer, a typical response is “no idea.” In fact, their solution process, though leading to the correct answer is due to Einstellung effect. An accurate, Einstellung-free, application of the reification principle ∙ 1, ∙ 2 1 and ∙3 yields , whence

2∙1

4.

146 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 6.7. Using reification principle in finding area of ABCD.

Fig. 6.8. Triangle ABC defines its four parts.

Multiple ways of finding areas using different utilizations of the reification principle may reduce the possibility of Einstellung effect. For ∙ example, using formula for the area of a triangle, the areas 1∙ 1

and

∙1∙

can be found; using ∙

formula for the area of a trapezoid, the areas 1

∙1

and

∙

1

1

∙1

can be

found; finally, area of quadrilateral ABCD shown in Fig. 6.7 can be 4. found as the sum Likewise, by incorrectly labeling each of the four parts (a, b, c, d) of triangle ABC in Fig. 6.8 as 1/2, one gets the correct value for area of the triangle. In the context of the discussion of Einstellung effect as related

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to a link between basic geometric principles and Gestalt psychology, the following formulation by Wertheimer [1938] of the essence of Gestalt theory is worth citing: “There are wholes, the behavior of which is not determined by that of their individual elements, but where the partprocesses are themselves determined by the intrinsic nature of the whole” (p. 2). Indeed, in Fig. 6.8 triangle ABC may be considered as a whole; however, it is not defined by its four parts (a, b, c, and d) – one triangle and three trapezoids – but rather the parts are defined by the whole triangle using the similarity of triangles. Because the ratio of the legs of the whole is 4 to 1, so is the ratio of the legs of triangle a (a part of the whole). This yields 1/4 as the length of the smaller leg of triangle a thus demonstrating that the triangular part of the whole is 1/8 of the unit square. Likewise, the smaller leg of the triangle comprised of a and b is equal to 1/2 whence it is one-half of the unit square. Consequently, trapezoid b is equal to 1/2 – 1/8 = 3/8 of the unit square. Trapezoid c is equal to 1 – 3/8 = 5/8 of the unit square and trapezoid d is equal to 7/8 of the unit square. It is not surprising that Gestalt psychologists used problem-solving strategies of informal geometry in teaching mathematics to young children (e.g., [Luchins and Luchins, 1970]). Their pedagogy was grounded in the tenets, “A whole is meaningful when concrete mutual dependency obtains among its parts” [Wertheimer, 1938, p. 16] and “the characteristics of the whole determine the characteristics of each part and its function in the whole” [Wertheimer, 1985, p. 23]. In particular, considering in Fig. 6.8 the rectangle comprised of four unit squares as a whole with eight parts, one can see how these parts depend on each other and their mutual dependency is due to the relation between the side lengths of the whole. An obvious connection between Gestalt theory and school geometry can be used as a motivation for posing new problems that go beyond informal geometry and require insight. To this end, the geometric construction of Fig. 6.8 can be extended to any number of unit squares cut by a diagonal connecting the two vertices of the far-right and the farleft unit squares. Furthermore, comparing the diagrams of Fig. 6.8 and Fig. 6.2 may prompt the idea of connecting the contexts of counting and geometry, and in doing so, to pose new problems. To this end, the

148 Integrating Computers and Problem Posing in Mathematics Teacher Education

diagram of Fig. 6.8 can be extended to any number of unit squares cut by a diagonal connecting vertices of the far-right and the far-left unit squares. In this general case, the following problems can be posed. Problem 6.5.1. What is the total number of segments serving as borders of the triangle and trapezoids in the construction comprised of n unit squares? Problem 6.5.2. Can the larger lateral side of any trapezoid so constructed be a rational number? Why or why not? Problem 6.5.3. Is there a trapezoid among this series of trapezoids into which a circle can be inscribed? (see [Hoehn, 1993, p. 285]). Why or why not? Problem 6.5.4. Is there a trapezoid among this series of trapezoids about which a circle can be circumscribed? Why or why not? The list of questions and problems can be extended. As Halmos [1980] noted, “I do believe that problems are the heart of mathematics, and I hope that as teachers, in classroom, in seminars, and in the books and articles we write, we will emphasize them more and more, and that we will train our students to be better problem-posers and problemsolvers than we are” (p. 524). This emphasis on the variety of educational means used to engage students in doing mathematics aiming at preparing a new generation of professionals who are endowed with higher quality mathematical skills in comparison with the previous generation implies using more and more problems that require productive thinking towards the end of reducing Einstellung effect in mathematical problem solving. Remark 6.1. Geoboard activities described in this section can be carried out using The Geometer’s Sketchpad (or any other dynamic geometry software) as a computational geoboard. Consequently, using such a digital tool makes it possible to find areas computationally. This alternative may serve as a verification of the correctness of using formal

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mathematics. In the case when the results of calculations are decimal fractions, one has to convert a decimal fraction into a common fraction or vice versa. 6.6 Einstellung effect in solving algebraic equations and inequalities Consider an algebraic equation of the form .

(6.1)

Often, the mindset developed through simplifying an equation (or even an equality) by cancelling out a common numeric factor in its both sides prompts the automatic usage of this simplification practice for equation (6.1). Considering as such a factor, one reduces equation (6.1) to the form g(x) = h(x)

(6.2)

and continues solving (6.2) as a simpler version of (6.1). However, when there exists such that 0 and , equation (6.1) is not equivalent to equation (6.2) as cancelling out results in the loss of as a root of equation (6.1). In that case, one can observe Einstellung effect in a sense that the strategy of cancellation of a common numeric factor when mechanically transferred to the equations of type (6.1) may lead to the loss of solutions. For example, consider the equation 2 which has two roots: x = 0 and x = 2. However, if one cancels x from both sides of the last equation, the equality x = 2 results and the root x = 0 gets lost. Likewise, when cancelling x as the common factor of both sides of the inequality 2 , one gets the inequality x > 2 as the solution. However, the 2 2 0 former inequality can be rewritten in the form from where it follows that either both factors are positive: x > 0 and x > 2, or both are negative: x < 0 and x < 2. That is, the solution consists of two disjoint intervals, namely, x > 2 and x < 0. Once again, after cancelling x from both sides of the inequality 2 , the solution x < 0 gets lost because the cancellation is error-free only when x > 0.

150 Integrating Computers and Problem Posing in Mathematics Teacher Education

Remark 6.2. Consider the equation 3 3 0. Its left-hand side can be factored in two different ways. One way is to have 3 3 0, thus focusing on x – 3, which then might be cancelled 1 0. The last equation does not out as a common factor to get have real roots and cancelling out x – 3 neglects the only real root, x = 3, that the equation has. However, the right-hand side of the cubic equation can be factored in another way, namely, 1 3 1 0, where the sum 1 is a factor which may be cancelled out as 1 0. This yields the equation x – 3 = 0, whence x = 3 – the root which survived Einstellung effect. The same is true for the inequality 3 3 0, which when replaced by 3 1 0 yields x > 3 as the only solution set. A similar, yet conceptually more complicated outcome of Einstellung effect can be observed through replacing the equality sign by the inequality sign in (6.1) and (6.2). To this end, consider Problem 6.6.1. Solve the inequality 1

2 1

0.

(6.3)

The solution graph of inequality (6.3) is shown in the form of the two bars in the bottom part of Fig. 6.9. If, for the sake of simplification, one cancels out the common factor 1 – x in the left-hand side of (6.3) without paying attention to its variable sign, this cancellation leads to a more complex outcome than just contraction of a solution set. When both sides of inequality (6.3) are divided by 1 – x, the resulting inequality has the form 1 2 1 0 or 2 4 1 0 whence √

√

.

(6.4)

The solution graph of inequalities (6.4) is shown through the bar in the top part of Fig. 6.9. Comparing the graphs of the two solution sets indicates simultaneous contraction and extension of the solution set of inequality (6.3) as an outcome of cancelling out the factor 1 – x (which is positive when x < 1 and negative when x > 1) for the purpose of “simplification”.

Einstellung Effect and Problem Posing

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In order to explain this complex phenomenon, note that the correct way of solving inequality (6.3) is to transform its left-hand side as follows: 1

1

1 1 2

2 1

1 4

4 1 ,

2

thus replacing (6.3) by the inequality 1 2

4

1

0.

(6.5)

Because the product of two factors is positive when the factors are of the same sign, the following two systems of inequalities (that distinguish between the cases x > 1 and x < 1) result from (6.5): 1

0, 2

4

1

0

(6.6)

1

0, 2

4

1

0.

(6.7)

and

One can see that the quadratic inequality in (6.7), the solution of which is presented through inequalities (6.4), has to be adjusted to satisfy the linear inequality in (6.7), that is, the inequality x < 1. Therefore, √

reducing (6.4) to

1 and comparing the latter to the solution

graph of (6.4), indicates that the part 1

√

does not belong to

the solution set of inequality (6.3). In other words, the cancellation of the factor 1 – x from both sides of inequality (6.3) resulted in the extension of its solution set. At the same time, inequalities (6.6), if their solution set is not empty, provide additional solution set, thus indicating a possible loss of solutions. Indeed, the quadratic inequality in (6.6) has the √

solution have

√

,

√

; due to the inequality x > 1 in (6.6) we

. It is the latter part of the x-axis that gets lost through the

erroneous cancellation of the factor 1 – x in inequality (6.3).

152 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 6.9. Solution set of (6.3) and (6.4) – bottom and top segments, respectively.

Remark 6.3. Replacing the inequality sign in (6.3) by the equality sign 0. Cancelling out 1 – x yields yields the equation 1 2 1 the quadratic equation 2

4

1

√

0 with the roots

.

Unlike the case of inequality (6.3), Einstellung effect in solving the above cubic equation leads to the loss of the root x = 1 and no extraneous root appears. That is, simultaneous contraction and extension of solution set caused by the cancellation of a common (variable) factor appears to be associated with an inequality only. Notwithstanding, there are cases when Einstellung effect in solving an equation can provide simultaneous contraction and extension of its solution set as well. For example, when solving the rational equation of 0, the transformation of the cubic polynomial

the form yields

0 or



0.

Dividing both sides of the last equation by the fractional factor results in the quadratic equation 5 6 0 with the roots x = 2 and x = 3. One can see that x = 2 is an extraneous solution to the original rational equation in which 2 0; at the same time, the root x = 1 has been lost. That is, Einstellung effect in solving both equations and inequalities can be responsible for simultaneous contraction and extension of solution set.

Einstellung Effect and Problem Posing

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Fig. 6.10. Solution set of (6.3) and (6.4) along with the graphs y = 1 – x and y = 2(1 – x)2.

Remark 6.4. Similar to the constructions demonstrated in Chapter 5, in the construction of the solution graphs the computer program Graphing Calculator, capable of graphing relations from any two-variable equations/inequalities, was used. For example, the solution graph of inequality (6.3) results from graphing simultaneously with (6.3) the inequality | | , where is a sufficiently small positive number which makes the solution graph of thickness 2 . Likewise, the solution graph of the inequality that resulted from (6.3) by cancelling out 1 – x can be constructed by graphing the inequalities 1 2 1 0 and simultaneously, where the numbers 0 make the solution graph of thickness . 6.7 Einstellung effect in solving trigonometric inequalities An example similar to algebraic inequality (6.3) can be provided in the case of a trigonometric inequality. With this in mind, consider

154 Integrating Computers and Problem Posing in Mathematics Teacher Education

Problem 6.6.2. Solve the inequality sin 2x < sin x

(6.8)

on the segment 0, 2 . Because sin 2 2 sin ∙ cos , one might (for the sake of “simplification”) reduce inequality (6.8) to that of 2 cos

1

(6.9)

by cancelling out sinx as the common factor in both sides of inequality (6.8). Comparing solution sets of inequalities (6.8) and (6.9) on the segment 0, 2 reveals a familiar phenomenon: the simultaneous contraction and extension of the solution set of inequality (6.8). This complex outcome can be demonstrated through the use of technology also. As shown in Fig. 6.11, the bottom bars represent the graph of the solution set of inequality (6.8) and the upper bar represents the graph of the solution set of inequality (6.9), both on the segment 0, 2 . In particular, the solution graph of inequality (6.8) was constructed from the system of inequalities sin sin 2 0, | | 0.05, 0 2 and the solution graph of inequality (6.9) was constructed from the system of inequalities 1 2 cos 0, 0.5 0.6, 0 2 . By graphing the solution sets along with the graphs of the functions y = sinx and y = 1 – 2cosx, one can see (Fig. 6.12) that cancelling out sinx from both sides of inequality (6.8) has led to the extension of its solution set (i.e., the space between the two bottom bars) when sin 0, 1 2 cos 0, and the contraction of its solution set (i.e., the loss of the bottom bar on the right) when sin 0, 1 2 cos 0.

Fig. 6.11. Simultaneous extension and reduction of a solution set.

Einstellung Effect and Problem Posing

155

Fig. 6.12. Graphing solution sets along with the graphs of the corresponding functions.

In order to explain the phenomenon of simultaneous extension and contraction of the solution set of inequality (6.8), analysis of graphs shown in Fig. 6.12 can be of help. In both cases (extension and contraction), the equivalence of solution sets when replacing (6.8) by (6.9) is violated when sinx < 0. Recall that the inequality ab < ac is equivalent to b < c when a > 0 and to b > c when a < 0. Indeed, the inequality ab < ac is equivalent to a(c – b) > 0 and the product of two numbers is positive when either both are positive: a > 0 and b < c, or both are negative: a < 0 and b > c. That is, if sin x > 0 (i.e., when 0 < x < then inequality (6.8) is equivalent to (6.9). This statement is confirmed by the graphs pictured in Fig. 6.12: when sinx > 0 the solution sets of (6.8) and (6.9) coincide on [0, 2]. When sinx < 0, inequality (6.8) is equivalent to the negation of inequality (6.9): 2cosx > 1 or 1 – 2cosx < 0; that is, solving inequality (6.9) on [0, 2] provides (6.8) with an extraneous solution. Furthermore, the solution to the negation of inequality (6.9) on [0, 2] gets lost. That is, on the one hand, when sin x < 0 inequality (6.9) is in error (providing extension of solution set); on the other hand, the (correct) negation of inequality (6.9) was never considered (thus the contraction outcome). This shows how computer graphing, capable of plotting solutions to two-

156 Integrating Computers and Problem Posing in Mathematics Teacher Education

variable inequalities, provides conceptual explanation of Einstellung effect. Remark 6.5. Replacing the inequality sign in (6.8) by the equality sign yields the equation sin2 sin . Solving this equation by cancelling out sin as a common factor from its both sides yields 2cos 1 with two ∈ 0, and ∈ , 2 on the segment [0, 2π]. At roots, the same time, the roots 0, , 2 of sin on [0, 2π] have been lost in the course of cancellation of sin . Therefore, Einstellung effect in the case of replacing inequality (6.8) by the corresponding equation is responsible for the contraction of solution set only. At the same time, similarly to the case of the algebraic equation mentioned in Remark 6.3, solving the equation 2tan 0 by replacing it with (

0 and then cancelling out the fractional factor

2tan ) yields the equation 1 sin 0 which has only one root on the segment [0, 2π]. Therefore, the cancellation of 2tan led

to the loss of the roots 0, , 2 and the emergence of for which tanx is undefined. extraneous root 6.8 Using technology to pose problems that might lead to Einstellung effect How can one use technology to pose problems on solving inequalities towards the goal of demonstrating a possible outcome of Einstellung effect – simultaneous expansion and contraction of solution set through cancelling out common factor from both sides of an inequality? The Graphing Calculator provides support for posing problems of that kind. To this end, one can define two functions, and , and then graph the solution sets of and 1 through the following definitions, respectively, 0, | | and

(6.10)

Einstellung Effect and Problem Posing

1

0,

157

(6.11)

0. In the case of a for some sufficiently small numbers 0, trigonometric inequality, in order to graph the corresponding solution set within the domain of periodicity of the functions involved, inequalities defining those domains in terms of x should be added. This may require one to determine the common period of the trigonometric functions involved. For example, in the case of inequality (6.8), the function sin has period 2π and the function sin2 has period π; therefore, 2π is their common period. Then, by trial and error (informed by a certain level of mathematical thinking) one can find appropriate expressions for the functions and in (6.10) so that the relation between the graphs of the two solution sets would look like those pictured in Fig. 6.11 and Fig. 6.12. For example, setting sin and 2cos in inequalities (6.10) and (6.11) leads to Problem 6.6.3. Solve the inequality sin

2sin ∙ cos

(6.12)

on the segment [0, 2 ] and explore what happens through reducing (6.12) to the inequality 1

2cos

.

(6.13)

The solution sets of inequalities (6.12) and (6.13) are shown in Fig. 6.13. Note that due to the identity 2cos 1 cos2 , the common period of the functions involved in (6.12) is 2π. One can see that when the common factor sin is cancelled out from both sides of inequality (6.12), the resulting inequality (6.13) has a solution set that manifests both the contraction and the extension of the solution set of (6.12). As shown in Fig. 6.13, the bottom bars represent the graph of the solution set of inequality (6.12) and the upper bars represent the graph of the solution set of inequality (6.13), both on the segment [0, 2 ] . Once

158 Integrating Computers and Problem Posing in Mathematics Teacher Education

again, this rather complex phenomenon can be demonstrated through the use of technology. Analyzing the graphs pictured in Fig. 6.14, one can conclude that the loss of solutions occurs when sin 0 and 1 2cos 0. At the same time, the extension of solution set of inequality (6.12) occurs when sin 0 and 1 2cos 0. To obtain error-free solution of inequality (6.12), one has to proceed 0 and factoring as follows. Reducing (6.12) to sin 2sin ∙ cos out sin yields sin 1 2cos 0 whence either both factors are positive, i.e., sin

0 and 1

2cos

0,

(6.14)

2cos

0.

(6.15)

or both factors are negative, i.e., sin

0 and 1

Fig. 6.13. Simultaneous extension and reduction of a solution set.

Fig. 6.14. Graphing solution sets along with the graphs of the corresponding functions.

Einstellung Effect and Problem Posing

159

As shown in Fig. 6.14, due to Einstellung effect, the extension of the solution set of (6.12) occurs when sin 0 and 1 2cos 0 – this pair of inequalities is not included in either (6.14) or (6.15). Furthermore, the transition from (6.12) to (6.13) is only true when sin 0; thus, we have the reduction of the solution set of inequality (6.12). Once again, everything is correct until the cancelled-out factor changes its sign from plus to minus. Remark 6.6. The use of technology allows one to establish conceptual sources of errors occurring due to Einstellung effect in solving inequalities. Knowing where errors may be coming from reduces the dependency of problem solvers on the mechanical transfer of experience without “insight into the structure but not superficial similarity of the new problem and the old problem whose solution is already understood” [Wertheimer, 1985, p. 24]. The appropriate use of modern technology and mathematical knowledge by teachers enables high school students to benefit from their teachers’ TPCK (technological pedagogical content knowledge) and, thereby, to become better problem solvers who are less dependent on the negative affordance of Einstellung effect. 6.9 Einstellung effect in solving logarithmic inequalities 6.9.1 Simultaneous extension and contraction of solution set Another type of functions included in the secondary mathematics curriculum is a logarithmic function log . When the logarithm’s base a = 10, the notation log is often used. A similar situation associated with Einstellung effect when solving logarithmic inequalities can be constructed using computer graphing software. Graphing inequalities (6.10) and (6.11) after setting log and 2 log 2 , results in the diagram of Fig. 6.15 where the bars penetrated by the x-axis represent solution sets of the inequality log

2 log 2 ∙ log

(6.16)

160 Integrating Computers and Problem Posing in Mathematics Teacher Education

and the bars at the top part of the diagram represent the solution set of the inequality 1

2 log 2 .

(6.17)

Once again, like in the case of inequalities (6.3), (6.8) and (6.12), comparing the graphs of solution sets of logarithmic inequalities (6.16) and (6.17) allows one to see a familiar phenomenon – simultaneous contraction and extension of the solution set of an inequality is due to Einstellung effect caused by cancelling out common factor, log this time. A useful practice in analyzing the source of Einstellung effect is to solve the so-constructed inequality (6.16) in a correct way, without cancelling common factor, and see what would happen.

Fig. 6.15. Graphing solution sets of inequalities (6.16) and (6.17).

Problem 6.9.1. Solve inequality (6.16) without transformations that may lead to Einstellung effect. The correct solution of inequality (6.16) requires one to carry out several actions which will be presented step by step. The first step deals with reducing inequality (6.16) to an equivalent combination of two systems of inequalities necessary for avoiding the cancellation of logx as a common factor. To this end, one starts with replacing (6.16) with its (obviously) equivalent form (by relating a product of two-variable factors to zero)

Einstellung Effect and Problem Posing

log

1

2 log 2

161

0

(6.18)

from where it follows that either log

0 and 2 log 2

1

(6.19)

log

0 and 2 log 2

1.

(6.20)

or

The second step is to solution set to each of log 0 yields x > 1/2 or yields log 2

solve inequalities (6.19) by finding a common the two inequalities. Solving the inequality 1 1. Solving the inequality 2 log 2 |log2 | 1/√2 whence 1/√2 log2

1/√2 . The last two inequalities can be replaced by log10

log2

√

log10√ whence, due to the the property of logarithmic function base 10 to increase monotonically, 10

2

√

10√ √

Note that from the inequality 1 ∙ √

between 1 and √

smaller than

√

√

or

.

an obvious relation

√

follows. What is less clear is whether 1 is greater or

. Thus, in the process of solving one problem, another

problem emerged. √

Comparing

to 1 constitutes the third step required to complete

the second step. Let us show (without using a calculator, yet using the fact that the exponential function increases monotonically when a > √

1) that 1

, that is, 10√

√2 we have 10

10√

10



√

9

2√ and 10√

2

show that 1

2. Indeed, because 10 > 4 = 22 and 2 2√

√

2. Another way to

is to note that the inequality 3

√



implies

2. So, the solution to inequalities (6.19) is

162 Integrating Computers and Problem Posing in Mathematics Teacher Education √

1

. One can see this solution graph in Fig. 6.15 as the bar on

the right-hand side of the x-axis. The fourth step is to solve inequalities (6.20). To this end, one starts with the inequality log 0 which is equivalent to 0 1. The inequality 2 log 2 1 leads to the inequality |log2 | whence √

log2

or log2

√

respectively, √

√

√

. The last two inequalities yield,

(not consistent with the inequality x < 1) and 0

. Because, 10√

2 we have

1 and, therefore, ∙

0

√

√

is the solution to inequalities (6.20).

The fifth step is to combine solutions to (6.19) and (6.20) and analyze the effect of Einstellung shown in Fig. 6.15. Such combination √

results in the unity of the inequalities 0

and 1

√

representing the correct solution set of inequality (6.16). However, when (6.16) was (incorrectly) replaced by (6.17), the solution 0

√

(adjacent to the origin as shown in Fig. 6.15) was lost due to Einstellung effect. By the same token, the upper bar in Fig. 6.15 as the solution set to 1 thereby (6.17) includes the extraneous solution ∙

√

demonstrating the extension of solution set as the result of Einstellung effect. 6.9.2 Extension of solution set Computational experiment supported by computer graphing software makes it possible to discover that depending on whether the value of n in the inequality log

log

∙ log

(6.21)

is even or odd, different manifestations of Einstellung effect result from cancelling out logx. To illustrate, consider the case n = 3 in (6.21) that

Einstellung Effect and Problem Posing

163

can also be presented as a step-by-step demonstration. The first step consists in “simplifying” the inequality log

3 log 3 ∙ log

(6.22)

by (erroneously) cancelling out logx from its both sides. This yields the inequality 1

3 log 3 .

(6.23)

Solving inequality (6.23) represents the second step of the demonstration. It begins with dividing both sides of (6.23) by 3 (this is permissible as 3 > 0), represent 1/3 as 1/3 / and factorize the difference of two cubes, log3 log3

, to get the inequality log3

log3

0.

Because the second factor in the left-hand side of the last inequality is positive for all x where log3 is defined (i.e., for x > 0), it is equivalent 0 and thereby the cancellation of a

to the inequality log3

variable expression which is positive on its domain of definition may not result in Einstellung effect. The last inequality can be re-written in the form log3

log10

whence 0

∙ 10

1.645.

(6.24)

The solution graph of inequalities (6.24), which are equivalent to (6.23), is shown at the top of the diagram of Fig. 6.16.

164 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 6.16. Einstellung effect leading to the extension of solution set only.

The third step is to solve inequality (6.22) without Einstellung effect. To this end, one has to replace (6.22) by the inequality log 1 3 log 3 0 which holds true when either log

0 and 1

3 log 3 ,

(6.25)

log

0 and 1

3 log 3 .

(6.26)

or

Solving (6.25) results in x > 1 and 0

∙ 10

1.645

whence the inequalities 1

∙ 10

(6.27)

define the solution set of (6.25). Solving (6.26) yields the inequalities 0

1 and

∙ 10

1.645 which are inconsistent. Therefore,

inequalities (6.27), the solution graph of which is shown at the bottom of the diagram of Fig. 6.16, define the solution set of inequality (6.22). One can see that transition from (6.22) to (6.23) results in Einstellung effect manifesting the extension of the solution set of the former inequality.

Einstellung Effect and Problem Posing

165

Remark 6.7. Using computer graphing software, one can check to see that whereas for even values of n in (6.21), cancelling the common factor logx results in simultaneous extension and contraction of solution set (e.g., Fig. 6.17), for odd values of n in (6.21) the Einstellung effect yields the extension of solution set only (e.g., Fig. 6.18). The graphs of the functions log and 1 2 log 2 demonstrate (Fig. 6.17) that when both log 0 and 1 2 log 2 0, the solution set of (6.17) includes that of (6.21) for n = 2 and the former includes the extension of the latter when log 0. At the same time, when both log 0 and 1 2 log 2 0, the solution set of (6.17) does not include that of (6.21) for n = 2. That is, neglecting the sign of log causes Einstelung effect in solving (6.21) for n = 2 that results in simultaneous extension and contraction of its solution set. Likewise, in the case n = 3, as demonstrated by the graphs of the functions log and 1 3 log 3 shown in Fig. 6.18, when the two functions assume positive values, the solution set of (6.23) includes that of (6.22); yet, when the function log assumes negative values, the solution set of (6.23) extends that of (6.22). At the same time, as the function log assume positive values for all x > 1, the contraction of the solution set of (6.22) through the transition to (6.23) does not occur. In the next section, this experimental observation will be explained analytically for n = 2k and n = 2k + 1. Yet, this explanation would require one to reason beyond distinguishing between even and odd values of n.

Fig. 6.17. The case n = 2 leads to extension and contraction for (6.21).

166 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 6.18. The case n = 3 leads to extension only for (6.21).

6.10 Solving logarithmic inequality (6.21) in the general case 6.10.1 The case n = 2k Consider the inequality log

2 log

2

∙ log

(6.28)

which is a special case of inequality (6.21) when n is an even number. The goal of the following demonstration is to develop a solution of (6.28) free from Einstellung effect. As before, it will be a step-by-step process. The first step is to replace inequality (6.28) by an equivalent inequality log 1 2 log 2 0 which, in turn, is equivalent to the following two systems of inequalities: log

0, 1

2 log

2

0

(6.29)

log

0, 1

2 log

2

0.

(6.30)

and

The second step is to find the solution set common to both inequalities in (6.29). Let us begin with replacing log 0 by the

Einstellung Effect and Problem Posing

167

equivalent inequality x > 1. Its counterpart in (6.29) can be solved through the following equivalent transformations: log

|

, |log 2

2 1 2

log10

,

log 2

1 2

log 2

log10

10

2

10

, ,

,

and, finally, 10

10

.

(6.31)

Now, one has to make inequalities (6.31) consistent with the 10

inequality x > 1. The inequality

1 is obvious as its left, is smaller than the number 1

hand side, written as the fraction ∙

for all k ≥ 1. This means that the left part of inequalities (6.31) is immaterial because the inequality x > 1 implies the inequality 10

. That is, in order to have (6.31) consistent with x > 1 one has

to write 1

10

.

(6.32)

Less obvious is a relation between the right part of inequalities (6.31) and the number 1. In order to establish this relation, one has to verify whether (6.32) holds true for all k ≥ 1. For example, one can check to see that

10

k = 2 ( 10

1 for k = 1 (already verified in section 6.9.1) and 1.27 …

1 . At the same time, for k = 3 we have

168 Integrating Computers and Problem Posing in Mathematics Teacher Education

10

0.92 …

1. Therefore, the fact that inequalities (6.32) do not

hold true when k = 3 implies that inequalities (6.31) and x > 1 are inconsistent for that value of k. In order to continue solving inequalities (6.29), the third step (an auxiliary one) has to be carried out. To this end, one has to conjecture and then prove that the inequality 10

1

(6.33)

holds true for all k ≥ 3. This would imply that when k ≥ 3 inequalities (6.29) have an empty solution set. Towards this end, the proof of inequality (6.33) begins with replacing it by 10 2 . Then, taking logarithm base 10 of both sides of the last inequality (using the property of logarithmic function base ten to increase monotonically along with the log2 . Repeating this operation

increase of its argument) yields one

more

2 log log2

time

yields

log

log log2

or

– log2

. Therefore, one has to prove the inequality 2 log log2

log 2

0

(6.34)

assuming 3. Under this assumption, inequality (6.34) holds true because its left-hand side increases monotonically (this statement can be confirmed in various ways using technology) and therefore for k ≥ 3 we have 2 log log2

log 2

6 log log6

log 6

0.12 …

0 .

Alternatively, one can use Wolfram Alpha to check that inequality (6.34) holds true beginning from k = 3. Therefore, when k ≥ 3, inequalities (6.29) have an empty solution set. It is only when k = 1 and k = 2 that reducing inequality (6.28) to the

Einstellung Effect and Problem Posing

169

inequality 1 2 log 2 0, which, as was demonstrated above, is equivalent to (6.31), results in both the extension and contraction of the solution and 0,

set 10

of

(6.28)

by

the

intervals

10

, 1)

, respectively (see Fig. 6.20). However, noticing the

presence of the inequality log 0 in (6.29) allows one to conclude that only when k = 1 and k = 2 the inequalities (6.32) provide solution to (6.29). This completes the second step of our demonstration which was supported by the third step. The fourth step is to solve inequalities (6.30). Similar to the case (k = 1) of inequalities (6.20), two systems of inequalities 0 < x < 1, 0

10

(6.35)

and 10

0 < x < 1,

(6.36)

10

have to be considered. Because

1 for k ≥ 1,

the solution of (6.35) has the form 0

10

, k ≥ 1

(6.37)

(see the bar adjacent to the origin in Fig. 6. 19 and Fig. 6. 20). Regarding the system of inequalities (6.36), taking into account inequality (6.33), one can conclude that when k ≥ 3, the inequalities 10 provide the solution to (6.36).

1

(6.38)

170 Integrating Computers and Problem Posing in Mathematics Teacher Education

10

At the same time, when k = 1 and k = 2 the inequality

1 holds true and, therefore, the solution set of inequalities (6.36) is empty. That is, only inequalities (6.37) provide the solution set to inequalities (6.30) and, consequently, to inequality (6.28) for all k ≥ 1. At the same time, when k = 1 the system of inequalities (6.29) provides an additional solution to inequality (6.28) in the form of inequalities (6.32). (The same is true when k = 2). This is in full agreement with the solution of inequality (6.16) – a special case (k = 1) of (6.28) – presented at the conclusion of section 6.9.1. The fifth step is to formulate in terms of parameter k the ultimate solution to logarithmic inequality (6.28) by combining solutions to (6.29) and (6.30) developed through the previous steps. First, inequalities (6.37) provide a solution set to (6.28) for all k ≥ 1. Second, inequalities (6.38) provide a solution set to (6.28) for all k ≥ 3. Finally, inequalities (6.32) provide a solution set to (6.28) for k = 1 and k = 2 only. Remark 6.8. The inequality 10

1 is rather imprecise (in fact, the

number 1 may be replaced by the number 2.5) and thereby, it does not allow one to show without a calculator that 10

1. Indeed, using

the former inequality would not allow one to prove the latter one as the following chain of relations demonstrates 10

∙ 10

∙ 10

∙1

.

That is why, in order to avoid artificially created difficulties, in full agreement with the TITE problem-solving concept, formal reasoning may be integrated with the use of a calculator. Remark 6.9. Solving inequality (6.28) by cancelling out logx as if it is positive for all x > 0, would always (i.e., for all k ≥ 1) result in the contraction of its solution set by the interval (0,

10

, which, as

Einstellung Effect and Problem Posing

171

shown in Fig. 6.19 and Fig. 6.20, is adjacent to the origin. The extension of solution set depends on the relation between number 1. If 10

and the

1, then, as shown in Fig. 6.20, the inequalities 1 represent an extraneous solution (the extension of

solution set). If 10

10

10

10

1, then the inequalities

10

represent an extraneous solution (the extension of solution

set). In any case, the simultaneous contraction and extension of solution set of inequality (6.28) is the outcome of Einstellung effect caused by dividing both sides of the inequality by logx. Mediating formal mathematical presentation by the use of technology brings about not only clarity to the presentation but also, in the spirit of the dictum “check the result” [Pólya, 1957, p. 59], such mediation serves as a visual verification of abstractness of formal mathematical reasoning through the concreteness of special cases.

Fig. 6.19. The case

10

1 (k ≥ 3).

172 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 6.20. The case

10

1 (k = 1, 2).

6.10.2 The case n = 2k + 1 Finally, consider the inequality log

2

1 log

2

1

∙ log

(6.39)

which is a special case of inequality (6.21) when n is an odd number. In turn, a special case (k = 1) of inequality (6.39) is inequality (6.22). The goal now is to develop (step by step) a solution of inequality (6.39) free from Einstellung effect. The first step is to divide both sides of inequality (6.39) by a positive number 2k + 1 and then, as before, reduce the resulting inequality to two pairs of simultaneous inequalities log

0,

log

2

1

0

(6.40)

log

0,

log

2

1

0.

(6.41)

and

The second step is to solve inequalities (6.40) which can be rewritten (after factorizing the left-hand side of the second inequality) as follows:

Einstellung Effect and Problem Posing

1, where

log 2

173

1

0

is the sum of products of the powers of log 2 . Because log 2

1

(6.42) 1

and

0 when x > 1, so is

.

Therefore, the second inequality in (6.42) is equivalent to the inequality log 2

1

0 which can be re-written in the form

. That is, inequalities (6.42) are log 2 1 log 10 equivalent to the pair of inequalities 0

∙ 10

,

1.

(6.43)

As before, the consistency of inequalities (6.43) and, consequently, the outcome of solving inequalities (6.40), depend on the relationship between

∙ 10

and the number 1.

So, the third step is to formulate in terms of k this critical relationship as a way of completing the second step. One can verify ∙ 10

numerically that for k ≤ 2 the inequality

1 holds

true. In that case, the solution set of (6.40) is not empty and would have the form 1

∙ 10

, k ≤ 2.

(6.44)

Comparing (6.43) to (6.44) indicates that neglecting the signs of logx and P(x) in solving (6.39) for k ≤ 2 would result in the extension of its solution set by the interval (0, 1]. By the same token, one can first conjecture and then prove that the inequality ∙ 10

1

(6.45)

174 Integrating Computers and Problem Posing in Mathematics Teacher Education

holds true for all k ≥ 3. Inequality (6.45) can be proved similar to inequality (6.33). One can conclude that when k ≥ 3, inequalities (6.43) and, consequently, (6.40) are inconsistent. The fourth step is to solve inequalities (6.41). When k ≤ 2, the second inequality

in

(6.41)

has

the

∙ 10

solution

;

consequently, x > 1. That is, when k ≤ 2 inequalities (6.41) are ∙ 10

inconsistent. When k ≥ 3, we have the inequalities 0 < x < 1 and ∙ 10

1 . Therefore,

∙ 10

have the solution 1.

(6.46)

Finally, the fifth step is to formulate in terms of k the solution set of logarithmic inequality (6.39) by combining solutions of inequalities (6.40) and (6.41). The following result can be formulated: inequalities (6.46) provide a solution set to (6.39) for all k ≥ 3; inequalities (6.44) provide a solution set to (6.39) for k = 1 and k = 2 only. This completes analytic demonstration of the case n = 2k + 1. This demonstration is in full agreement with the solution of inequality (6.22) which is a special case (k = 1) of (6.39). Indeed, setting k = 1 in (6.44) yields (6.27). Remark 6.10. Using Wolfram Alpha in solving inequalities like (6.21) depending on parameter may not be feasible unless its Alpha Pro version is used. That is why, the introduction of parameter leads to TITE problems which are technology-immune because even the final answer cannot be found. Yet, an inequality with parameter is still technologyenabled in the presence of the Graphing Calculator capable of using a parameter-loaded scroll bar to obtain a numeric approximation to the solution for a particular value of parameter. Remark 6.11. Knowing conceptual characteristics of problem-solving techniques for inequalities involving logarithmic functions makes it possible to formulate various inequalities with non-linear expressions

Einstellung Effect and Problem Posing

175

under the logarithm. For example, such a problem could have the form of the inequality log

3

4 ∙ log

2

∙ log

3

(6.47)

which may first be modeled using computer graphing software in much the same way as it was demonstrated above. Having the results of modeling in the form of a graph of the solution set can facilitate formal problem-solving techniques. In that way, problems on solving algebraic inequalities can become TITE problems – technology is used to provide computational verification of results obtained through formal mathematical reasoning. For example, the expression 1 1 √10 will be a part of formal solution to inequality (6.47) as shown in Fig. 6.21 created by Wolfram Alpha. Yet, computational solution through the construction of solution graphs (Fig. 6.22), shown, respectively, as the bottom and the top bars, of the inequalities (6.47) and its “simplification” through cancelling out log 3 apparently causing Einstellung effect (both extension and contraction of solution set) provide the approximation -2.14727 to the radical expression. In order to make solving inequality (6.47) a TITE problem, the following problems can be posed. 1. Without using a calculator, show that 1

10

√10

√3.

Can this be demonstrated through a single squaring? 2. Why is the length of the part of the solution set of (6.47) located to the left of the origin smaller than one? 3. Why is the region to the left of the origin not affected by the “simplification” as the solution graphs of Fig. 6.22 demonstrate? 4. Replace in (6.47) the inequality sign by equality sign and solve the equation. Describe possible outcomes of Einstellung effect caused by the cancellation of the term log 3 . Compare solution set of the equality to that of inequality (6.47). Use Wolfram Alpha to verify your conclusion.

176 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 6.21. Wolfram Alpha solution to (6.47).

Fig. 6.22. Solution graphs of (6.47) and its “simplification”.

6.11 Summary This chapter investigated the relationship between problem posing and Einstellung effect. It started with examples of using proportional reasoning in conceptually distinct counting situations when this reasoning, otherwise known as being a successful counting strategy, is erroneous. At the same time, erroneous reasoning, when its source is explained conceptually, can be used as a thinking device in posing problems evaluating the effect of an error on the result of counting. A historical account of the studies of the relation between productive thinking and Einstellung effect was provided through the discussion of classic water jar experiments. In this context, a spreadsheet was used as a tool for posing problems in support of water jar experiments aimed at the remediation of Einstellung effect. Activities on a geoboard were connected to the notion of reification – one of the main pillars of Gestalt psychology, and typical errors made by elementary teacher candidates engaged in geoboard activities were analyzed. Einstellung effect and its

Einstellung Effect and Problem Posing

177

impact on solving rather advanced polynomial, trigonometric, and logarithmic equations and inequalities in one variable have been discussed in great detail. The use of the Graphing Calculator was suggested both as a means of posing equations and inequalities through solving of which consequential errors are likely to occur and as a conceptual analyzer of the sources of errors due to Einstellung effect. The next chapter concerns posing and solving TITE problems about patterns formed by the last digits of various integer sequences, including subsequences of polygonal numbers and that of the sums of the powers of integers.

b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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Chapter 7

Explorations with Integer Sequences as TITE Problem Posing 7.1 Introduction In this chapter, explorations with different integer sequences will be considered under the umbrella of TITE problem posing. Integer sequences are all subsequences of the natural number sequence. But unlike the sequence of natural numbers in which the rule that governs its development is clear – add one to the previous number to get the next term of the sequence – most of the integer sequences encountered by mathematicians and others do not have an obvious rule in order to determine the next term of a sequence. As was mentioned in Chapter 3, section 3.4, there exists the Online Encyclopedia of Integer Sequences (OEIS®, https://oeis.org), a rich source of mathematical knowledge. The OEIS®, grown out of the Handbook of Integer Sequences [Sloane, 1973], enjoys more than 10,000 downloads per day. Integer sequences appear in different contexts, both in mathematics and elsewhere. They include contexts as diverse as the growth of the population of rabbits breeding in ideal circumstances described by one of the most famous in mathematics Fibonacci27 number sequence 1, 1, 2, 3, 5, 8, 13, ..., and that of the sequence 10, 300, 9405, 271701, 7586055, … representing the number of ways to construct a triangle with the longest side n using unit-length straws of three colors for the sides [Abramovich, 2016c], just recently included in the OEIS® (sequence A278037). For example, when n = 1 one has to select three straws out of three color types allowing for the ! repetition of colors, something that can be done in 10 ways (see ! ∙ !

Chapter 5, section 5.2). The material of this chapter is motivated by a rather standard problem from an advanced secondary school mathematics curriculum. In 27

Leonardo Fibonacci (1170–1250) – the most prominent Italian mathematician of his time, credited with the introduction of Hindu-Arabic number system into the Western World. 179

180 Integrating Computers and Problem Posing in Mathematics Teacher Education

that problem, the last digit of a certain sum of powers of integers is used as its characteristic. This prompted the author to consider integer sequences, which are subsequences of a number of well-known sequences, through the lenses of the last digit of their terms. Such perspective makes it possible to pose a variety of problems that can integrate user friendly digital tools with formal reasoning techniques needed to produce empirical evidence as a source of information and make sense of tool-motivated observations by offering grade-appropriate explanations. 7.2 Exploring patterns formed by the last digits of the sums of powers of integers In a problem-solving book geared towards secondary school students with special interest in mathematics, the following problem can be found: Compute the units digit of 15 + 25 + 35 + … + 20065 [Linker and Sultan, 2016, p. 6]. In the age of computers, the units digit (alternatively, the last digit) can be found computationally by simply calculating this sum (e.g., by typing in the input box of Wolfram Alpha the quest “sum of i^5 from i=1 to i=2006”) and then looking at its last digit which turns out to be the unity (Fig. 7.1). In that way, as the quest for calculating the sum is formulated in very basic terms, the answer to the problem becomes available almost at the push of a button. Therefore, a question to be explored is how one can turn this problem into a TITE one enabling the modern-day integration of argument and computation.

Fig. 7.1. Using Wolfram Alpha to find the sum.

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181

One direction is to pose several problems by varying the last integer in the sum of the fifth powers of integers. This, however, would not add much TITE flavor to the problem unless such a variation is the focus of explorations. With this in mind, the following problem can be posed. Problem 7.1. Consider the sum ∑ for different values of n. Develop a formula for calculating the last digit of an integer28. Explore whether the last digits of the sum of the fifth powers of integers form a pattern as n changes. Extend the exploration to the sums of other powers of integers. TITE scenario. The problem can be explored in a variety of ways using a variety of digital tools. For example, one can enter the command “Table[sum of i^5 from i = 1 to i = n, {n, 100}]” into the input box of Wolfram Alpha to get a list of 100 integers (Fig. 7.2): 1 , 1 2 ,1 2 3 , . .. , ∑ .

Fig. 7.2. Using Wolfram Alpha to generate 100 consecutive sums of the first fifth powers. 28

In this chapter, only positive integers will be considered.

182 Integrating Computers and Problem Posing in Mathematics Teacher Education

Note that the last digits of integer powers of natural numbers form at most a four cycle. Indeed, the last digits of the powers of 2, 3, 7, and 8 form (different) four-cycles (some of which are pretty similar) – {2, 4, 8, 6}, {3, 9, 7, 1}, {7, 9, 3, 1}, and {8, 4, 2, 6}, respectively; the last digits of the powers of 4 and 9 form (different) two-cycles – {4, 6} and {9, 1}, respectively; finally, the last digits of the powers of 1, 5, and 6 do not change at all as the exponent changes. While this observation implies that the last digits of the 5th powers of consecutive integers form cyclic patterns, it is not immediately clear that the last digits of the consecutive sums of the first 5th powers of integers form cycles. However, a brief look at the results of computing enables one to see that the last digits of ∑ the terms of the sequence ,5 form the cycle {1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0} of length 20. Missing are the digits 2, 4, 7, 9; the digits 0, 1, and 5 each appear four times. The same cycle can be observed for x(n, 10), x(n, 15), x(n, 20). This observation can be done in the context of Maple. Moreover, one can check to see that the sum of the elements of the cycle is a multiple of ten; that is, each sum of 20 consecutive fifth powers has zero as the last (alternatively, units) digit. This suggests an alternative solution to Problem 7.1 – because the last digit of the sum ∑ is zero (this conclusion may be a TE part of equals the last digit of the the problem), the last digit of the sum ∑ sum 2001 2002 ⋯ 2006 . The sum of the last digits of each of the six addends is equal to 1 + 2 + 3 + 4 + 5 + 6 = 21 (this conclusion may be considered a TI part of the problem). This confirms a purely computational result shown in Fig. 7.1. Given an integer N, how can one define its last digit, LD(N), through a formal mathematical definition? Answering this question is another TI component of the TITE scenario. For example, 9 LD 159 159 150 159 10 ∙ 15 159 10 ∙ INT , where (in the context of a spreadsheet) the notation INT(x) denotes the largest integer not greater than x. In general, the formula LD

10 ∙ INT

(7.1)

for any (positive) integer N returns its last digit. That is, regardless of N, formula (7.1) includes simple arithmetical operations between N and the

Explorations with Integer Sequences as TITE Problem Posing

183

number 10. Note that in the context of Maple, one has to use the notation floor(x) instead of INT(x). As shown in Fig. 7.3, Maple converts the notation floor(x) into . In order to prove formula (7.1), let 10 10 ⋯ 10 , an n-digit base ten integer in which ∈ 0, 9 , 0. Then 10

10 ∙ INT 10

10

10

⋯

⋯

10

LD

This completes the proof. If one defines the sequence of integers expression LD

,

,

10

10 ∙ INT

. ∑

, ,

, then the

would display the

last digits of the sequence , . In order to verify the existence of a 20-cycle, one can use Maple which can show that the values of are always the same for every ∈ and LD 20, LD , 1, 10 or even for a larger power of ten. Such Maple environment incorporates a loop with an if statement through which, given the value of k, the program checks whether an inequality takes place between for any value of n within a and LD 20, LD , sufficiently large range. As shown in Fig. 7.3 (an empty space below the last line of the code), when k = 5 no such inequality between LD , exists among the first 104 values of n. and LD 20,

Fig. 7.3. Using Maple showing a 20-cycle in the sequence of the last digits of the sum.

184 Integrating Computers and Problem Posing in Mathematics Teacher Education

The above observations prompt extending the exploration to the sums of other powers of integers and check whether a 20-cycle phenomenon observed among the last digits of the sums would be taking place. For example, the last digits of the terms of the sequence ,2 ∑ form the cycle {1, 5, 4, 0, 5, 1, 0, 4, 5, 5, 6, 0, 9, 5, 0, 6, 5, 9, 0, 0} of length 20, the sum of the elements of the cycle being a multiple of ten and with the digits 2, 3, 7, 8 missing. The same cycle formed by the last digits can be observed for the terms of the sequences x(n, 6), x(n, 10), and x(n, 14). Why is it so? Answering this question is a TI part of the TITE scenario. To this end, one can recall, as it was already mentioned above, that the last digits of the integer powers of the first ten natural numbers form at most a four cycle as the exponent changes. While this observation does not explain the formation of the cycle of length 20, it explains why the last digits of the terms of the sequences x(n, 2 + 4k) are the same for k = 0, 1, 2, … .

Fig. 7.4. The last digit of the sum ∑

is not zero.

In order to explain the 20-cycle phenomenon, note that if m is the smallest integer for which the last digit of the sum ∑ is zero,

Explorations with Integer Sequences as TITE Problem Posing

185

then the last digits of the sequence , form a cycle of length 10m. As shown in the chart of Fig. 7.4 developed within a spreadsheet (see Appendix for programming details) a 20-cycle does not occur for k = 4, 8, …, because the last digit of each of the sums ∑ and ∑ is not zero (cell AJ22). At the same time, as shown in Fig. 7.5, a 20-cycle does occur for k = 2, 6, …, because the last digit of each of the sums ∑ and ∑ is zero (cell Y22). By extending the spreadsheet exploration beyond the sum of the first 20 fourth powers, one can see (Fig. 7.6) that m = 10 is the smallest integer for which the last digit of the sum ∑ is zero. In addition, one can pose a new set of problems: find the last digits of the terms of the sequences x(n, 4k + 2) for n = 2002, 3003, 4004 and k = 1, 2, 3, … . All the problems require integration of argument and computation. The intriguing character of such explorations motivates other integer sequences to be explored. To this end, the next section will deal with polygonal numbers.

Fig. 7.5. The last digit of the sums ∑

and ∑

is zero.

186 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 7.6. The last digit of the sum ∑

is zero.

7.3 Discovering patters in the last digits of the polygonal numbers 7.3.1 The triangular number sieves In section 1.5.2 of Chapter 1, subsequences of triangular numbers called the triangular number sieves of order k, with the triangular numbers (i.e., the numbers 1, 3, 6, 10, 15, 21, 28, …) being considered the triangular number sieve of order zero, were introduced and hexagonal numbers (i.e., the numbers 1, 6, 15, 28, …) were referred to as the triangular number sieve of order one. In what follows, closed and recursive formulas for the triangular number sieves of order k will be developed

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187

and such developments will be extended to other polygonal numbers. Moreover, the polygonal number sieves will be explored in terms of the last digits of their terms and the corresponding explorations will be treated as TITE problems. Note that all the sequences resulting from successive elimination of every other term of the corresponding subsequence of triangular numbers are quadratic sequences of the rank of a number within a sieve. For example, and , 2 1 , so that, , 6 and , 15. Furthermore, as

,

,

2

,

1 ,

we have ,

2

,

1 2 2

,

,

,

1 4 8

,

1 3 8 7 16

2 1 4 7 , 15 ,

3 ,

and, in general, ,

2

2

1 2

2

1 .

(7.2)

Formula (7.2) is a closed-form representation of the triangular number sieve of order k. One can prove formula (7.2) using the method of mathematical induction by variable k. To this end, setting k = 0 in (7.2) yields the well-known formula for triangular numbers. Indeed, 1 1 1 1 1 1 1 1 . , 2 2 2 2 2 Next, assuming that formula (7.2) holds true, one can carry out the transition from k to k + 1 as follows: 2 1 2 2 1 2 1 2 1 2 , , 2 2 2 1 2 2 2 1 2 2 1 2 2 1 . Just as the triangular numbers , (alternatively, the triangular number sieve of order zero) can be represented through the recursive , , 1, the triangular number sieve of definition , , order k can be defined recursively. To develop such definition, one has to 1. Then, find the difference , , , keeping in mind that , formula (7.2) can be re-written as follows: ,

2

2

3∙2

2

3∙2

1. (7.3)

188 Integrating Computers and Problem Posing in Mathematics Teacher Education

Using formula (7.3) enables the following algebraic transformations: ,



2 2 2

2

2 2 3∙2 1 2 2 3∙2 3 ∙ 2 (2 1 . ,

3∙2

2

2

1

2 3∙2

3∙2

(2

3∙2 3∙2 3∙2

1 1

That is, ,



,

2

1 ,

,

1.

(7.4)

Formula (7.4) is a recursive definition of the triangular number sieve of order k. One can check to see that when k = 0 we have , , , , 1 – the recursive definition of triangular numbers; when k = 1 4 3, , 1. The latter definition produces we have , , the sequence of hexagonal numbers 1, 6, 15, 28, … – the triangular number sieve of order one. Note that checking results is one of the important components of problem posing – before offering a problem to students to solve, it not only has to be solved by a teacher, but, most importantly, the correctness of the solution has to be verified; e.g., through a recourse to special cases as it was done above. Checking the result is also an important means of providing rigor while practicing mathematics as an experimental science. 7.3.2 Triangular number sieves and the last digits of their terms In the context of investigating integers in terms of their digits, an interesting exploration is to explore patterns in the sequences of the last digits of the terms , . Experimentally, by using a spreadsheet and either (closed) formula (7.2) or (recursive) formula (7.4), one can discover that the sequence LD , forms a cycle of length 20; the sequence LD , forms a 10-cycle; the sequence LD , forms a 5-cycle and this last 2. property appears being invariant for the sequences , , Furthermore, the sequences LD , for 2 consist of 5-digit strings with two ones, two fives, and one three. Whereas the number of ! 30 (see permutations in the string [1, 1, 3, 5, 5] is equal to ! ∙ !

Explorations with Integer Sequences as TITE Problem Posing

189

Chapter 5, section 5.2), only four permutations form cycles for LD , , 2. These permutations have an interesting property, the introduction of which requires familiarity with an advanced, yet easy to understand, combinatorial concept. 7.3.3 Rises and falls in permutations It is said (e.g., [Comtet, 1974]) that the permutation induces a rise in the element ,1 , 1 , 2 , 3 ,…, if 1 . Likewise, it is said that the permutation induces a fall in the element ,1 , 1 , 2 , 3 ,…, if 1 . Consequently, it is said that the permutation has exactly n rises [falls] on the set 1 , 2 , 3 ,…, 1, 2, 3, … , if there exist exactly n – 1 values of i such that 1 1 . For example, the sequence LD , forms a cycle defined by the string [1, 5, 5, 1, 3] – a permutation with exactly three rises and two falls as in the chain of the inequalities 1 < 5 = 5 > 1 < 3 the signs “” occur exactly twice and once, respectively; the sequence LD , forms a cycle defined by the string [1, 5, 3, 5, 1] – a permutation with exactly three rises and three falls as in the chain of the inequalities 1 < 5 > 3 < 5 > 1 the signs “” both occur exactly twice; the sequence LD , forms a cycle defined by the string [1, 3, 1, 5, 5] – a permutation with exactly three rises and two falls as in the chain of the inequalities 1 < 3 > 1 < 5 = 5 the signs “” occur exactly twice and once, respectively; and the sequence LD , forms a cycle defined by the string [1, 1, 5, 3, 5] – a permutation with exactly three rises and two falls as in the chain of the inequalities 1 = 1 < 5 > 3 < 5 the signs “” occur exactly twice and once, respectively. Finally, the sequences LD , and LD , form identical permutations for k = 2, 3, 4, … . 7. 7.3.4 Connecting triangular and square numbers within the multiplication table Square numbers (alternatively, polygonal numbers of side four) represent a well-known number sequence appearing in the context of the

190 Integrating Computers and Problem Posing in Mathematics Teacher Education

multiplication table and residing on its main diagonal as the products of two equal factors. Triangular numbers, however, are not an explicit part of the multiplication table; explicitly they can be found comprising one of the diagonals of Pascal’s triangle (Chapter 1, section 1.2) – a structure which students encounter long after they are introduced to the multiplication table. Nonetheless, the two sequences are closely related: the sum of two consecutive triangular numbers is a square number. This fact (attributed to Theon29) is hidden in the multiplication table. An interesting activity is to uncover triangular numbers in the multiplication table – they are located at the end points of each step of the kind of a ladder as shown in Fig. 7.7. Pairs of consecutive triangular numbers of the same parity (i.e., 6 and 10) are located horizontally above the main diagonal of the multiplication table; pairs of consecutive triangular numbers of different parity (e.g., 21 and 28) are located vertically above the main diagonal of the table. The opposite is true for the pairs of same and different parities below the main diagonal of the table. The greater the elements of the pairs (located either horizontally or vertically), the larger the distance from their respective locations to the main diagonal of the multiplication table.

Fig. 7.7. Connecting triangular and square numbers in the multiplication table. 29

Theon – a 4th century Greek mathematician.

Explorations with Integer Sequences as TITE Problem Posing

191

7.3.5 The square number sieves Just as in the case of the triangular numbers, the square number sieves may be considered. The square number sieve of order zero is the sequence of square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100,… described by the closed formula .

,

(7.5)

Problem 7.2. Let , denote the n-th term of the subsequence of square numbers that survived k successive eliminations of the terms of even rank. Develop closed and recursive formulas for the sequence , . Use the method of mathematical induction to prove the former formula. Explore a pattern formed by the last digits of the sequence , for different values of k (that is, for the square number sieves of order one, two, three, etc.) TITE scenario. The scenario begins with its TI part which requires certain algebraic skills and experience in inductive generalization. To this end, by eliminating every square number of even rank from sequence (7.5) one gets the sequence 1, 9, 25, 49, 81, … described by the closed formula ,

2

,

1 .

(7.6)

Sequence (7.6) will be called the square number sieve of order one. By eliminating every number of even rank from sequence (7.6) one gets the sequence 1, 25, 81,… described by the closed formula ,

,

2 2

1

1

4

3 .

(7.7)

Sequence (7.7) will be called the square number sieve of order two. By eliminating every number of even rank from sequence (7.7) one gets the sequence defined by the formula ,

,

4 2

1

3

8

7 .

Sequence (7.8) will be called the square number sieve of order three.

(7.8)

192 Integrating Computers and Problem Posing in Mathematics Teacher Education

Taking notice of the identities 2 4

3

2 2

1 , 2 1 2 1 , 8 7

2

2 2

1 , 2 1 ,

makes it possible to generalize from formulas (7.5)–(7.8) to get a closed formula 2 2 1 . (7.9) , Formula (7.9) will be called the square number sieve of order k. To prove formula (7.9) using the method of mathematical induction a by the variable k, one can set k = 0 in formula (7.9) to get , true formula. Assuming that formula (7.9) is true, the transition from k to k + 1 can be carried out as follows:

,

,

2

2

2 2 1 .

1

2

1

2

2∙2

1

One can derive a recursive formula for the square number sieve of order k in the form

,

2 2

,

3∙2

2 ,

The transition from formula (7.9), in which (7.10) can be carried out by finding the difference ,

2 2

2 2 1 2 1 2 2 ,

,

2

1 3∙2

1 2

2 1 2

2 1

(7.10)

1, to formula

,

1 2

1.

,

1 2

1

2 ,

whence formula (7.10). One can set k = 0 in (7.10) to get the well-known 2 1, recursive formula for the square numbers: , , 1. This completes a TI part of the TITE scenario for Problem 7.2. , A TE part of the scenario requires one to generate the sequences LD , for different values of k. Using formula (7.1), the following results can be obtained within a spreadsheet. The results of this exploration will be formulated in combinatorial terms using the notion of rises/falls induced in the elements of a permutation. The sequence LD , forms the cycle 1, 9, 5, 9, 1 – a permutation with exactly three rises and three falls as in the chain of the inequalities 1 < 9 > 5 < 9 > 1 each of the signs “” occurs exactly twice. The

Explorations with Integer Sequences as TITE Problem Posing

193

sequence LD , forms the cycle 1, 5, 1, 9, 9 – a permutation with exactly three rises and two falls as in the chain of the inequalities 1 < 5 > 1 < 9 = 9 the signs “” occur exactly twice and once, respectively. The sequence LD , forms the cycle {1, 1, 9, 5, 9 – a permutation with exactly three rises and two falls as in the chain of the inequalities 1 = 1 < 9 > 5 < 9 the signs “” occur exactly twice and once, respectively. The sequence LD , forms the cycle 1, 9, 9, 1, 5 – a permutation with exactly three rises and two falls as in the chain of the inequalities 1 < 9 = 9 > 1 < 5 the signs “” occur exactly twice and once, respectively. The sequence LD , forms the cycle 1, 9, 5, 9, 1 – a permutation with exactly three rises and three falls as in the chain of the inequalities 1 < 9 > 5 < 9 > 1 each of the signs “” occurs exactly twice. The sequence LD , forms the cycle 1, 5, 1, 9, 9 – a permutation with exactly three rises and two falls as in the chain of the inequalities 1 < 5 > 1 < 9 = 9 the signs “” occur exactly twice and once, respectively. 7.3.6 The pentagonal number sieves The next step in this TITE exploration of subsequences of polygonal numbers is to develop the pentagonal number sieve of order k. To this end, consider the sequence of pentagonal numbers (alternatively, polygonal numbers of side five) 1, 5, 12, 22, 35, 51, 70, … defined by the closed formula ,

(7.11)

Problem 7.3. Let , denote the n-th term of the subsequence of pentagonal numbers that survived k successive eliminations of the terms of even rank. Develop closed and recursive formulas for the sequence , . Prove the former formula by the method of mathematical induction. Explore a pattern formed by the last digits of the sequence , for different values of k (that is, for the pentagonal number sieves of order one, two, three, etc.) TITE scenario. Eliminating every pentagonal number of even rank from sequence (7.11) yields the pentagonal number sieve of order one defined by the formula

194 Integrating Computers and Problem Posing in Mathematics Teacher Education ,

2

,

1 3

2 . (7.12)

Eliminating every number of even rank from sequence (7.12) yields the pentagonal number sieve of order two defined by the formula ,

2 2 5 .

,



4

3 6

1

1 3 2

1

2 (7.13)

Eliminating every number of even rank from sequence (7.13) yields the pentagonal number sieve of order three defined by the formula ,

4 2 11 .

,



8

7 12

1

3 6 2

1

5 (7.14)

Generalizing from formulas (7.11)–(7.14) yields the formula 2

,

2

1 3∙2

3∙2

1

(7.15)

which defines the pentagonal number sieve of order k. To prove formula (7.15) using the method of mathematical induction by the variable k, note that setting k = 0 in formula (7.15) yields formula (7.11), a closed 1 1 formula for pentagonal numbers. Indeed, , 1

. Transition from k to k + 1 can be carried out as follows:

2 2 1 3∙2 1 2 2 1 3∙ 1 2 2 1 3∙2 3∙2 1 . 2 In order to derive recursive formula for the pentagonal number sieve of order k, one has to find the difference , , . This can be a TE part of the scenario involving Wolfram Alpha. The program simplifies this difference defined through formula (7.15), in which , 1, to the 3∙2 2 3 5 so that the formula can be written as form 2 follows ,

,

,

,

2

3∙2 2

3

5,

,

1.

(7.16)

Explorations with Integer Sequences as TITE Problem Posing

195

Formula (7.16) is a recursive formula for the pentagonal number sieve of order k. Exploring patterns formed by the last digits of the sequences , within a spreadsheet, the following results can be obtained. The sequence LD , forms a 20-cycle. The sequence LD , forms a 10-cycle. The sequence LD , forms the cycle 1, 5, 7, 7, 5 which is a permutation with exactly three rises and two falls as the chain of the inequalities 1 < 5 < 7 = 7 > 5 includes the signs “” exactly twice and once, respectively. The sequence LD , forms the cycle 1, 7, 5, 5, 7 which is a permutation with exactly three rises and two falls as the chain of the inequalities 1 < 7 > 5 = 5 < 7 includes the signs “” exactly twice and once, respectively. The sequence LD , forms the cycle 1, 5, 7, 7, 5 which is a permutation with exactly three rises and two falls as the chain of the inequalities 1 < 5 < 7 = 7 > 5 includes the signs “” exactly twice and once, respectively. The sequence LD , forms the cycle 1, 7, 5, 5, 7 which is a permutation with exactly three rises and two falls as the chain of the inequalities 1 < 7 > 5 = 5 < 7 includes the signs “” exactly twice and once, respectively. The sequence LD , forms the cycle 1, 5, 7, 7, 5 which is a permutation with exactly three rises and two falls as the chain of the inequalities 1 < 5 < 7 = 7 > 5 includes the signs “” exactly twice and once, respectively. The sequence LD , forms the cycle 1, 7, 5, 5, 7 which is a permutation with exactly three rises and two falls as the chain of the inequalities 1 < 7 > 5 = 5 < 7 includes the signs “” exactly twice and once, respectively. 7.3.7 The general case of the m-gonal number sieves In this section, closed and recursive formulas for the sequence that will be referred to as the m-gonal number sieve of order k will be developed. To begin, note that a general closed formula for the polygonal numbers of side m (alternatively, m-gonals) has the form ,

2

.

(7.17)

196 Integrating Computers and Problem Posing in Mathematics Teacher Education

In particular, ,

,

1

,

,

∙3

,

.

Problem 7.4. Let , , denote the n-th term of the subsequence of mgonal numbers defined by formula (7.17) that survived k successive eliminations of the terms of even rank. Develop closed and recursive formulas for the sequence , , . Prove the former formula by the method of mathematical induction. Construct a spreadsheet that given n, m, and k, returns the last digit of the term , , . TITE scenario. In what follows, the polygonal number (of side m) sieve will be referred to as the m-gonal number sieve. We begin with a TI part of the scenario by eliminating every number of even rank from sequence (7.17) to get 2 1 2 2 2 2 1 , , , , 2 2 1 1 2 2 1 2 1 1 2 1. That is, the m-gonal number sieve of order one is represented by the formula 2

, ,

1

1

2

1.

(7.18)

Eliminating every number of even rank from sequence (7.18) yields 2 2 1 1 2 2 2 1 , , , , 1 2 1. 4 3 2 That is, the m-gonal number sieve of order two is represented by the formula 4

, ,

3 2

1

2

1.

(7.19)

Eliminating every number of even rank from sequence (7.19) yields , ,

, ,

8 2

4 2 1 3 2 1 2 7 2 2 1 2 1

1 1 2

2 1.

1

Explorations with Integer Sequences as TITE Problem Posing

197

That is, the m-gonal number sieve of order three is represented by the formula 2

, ,

1 2

2

1

2

1.

(7.20)

Generalizing from formulas (7.18)–(7.20) yields 2

, ,

2

1 2

1

2

1.

(7.21)

Formula (7.21), a closed formula for the m-gonal number sieve of order k, can be proved using the method of mathematical induction by the variable k. To this end, one can note that setting k = 0 in formula (7.21) should result in formula (7.17). Indeed, 1 2

, ,

1

2

1

1

2

2

,

.

Assuming that formula (7.21) is true, the transition from k to k + 1 can be carried out as follows: , ,

2 2 2

1 2

, ,

2

1 2 1 2

2 1

2

2 1.

2

1

This completes the mathematical induction proof of (7.21). 1, and taking advantage of Using formula (7.21), in which , , the power of symbolic computations provided by Wolfram Alpha, one can simplify the difference , , , , (as shown in Fig. 7.8) to get the recursive formula , ,

2

2

, ,

2 2

3

,

1. (7.22)

, ,

In particular, when m = 4 formula (7.22) yields formula (7.10). Indeed, ,

, ,

, , ,

2 2

2

2 4 3∙2

2 2 2 ,

3 ,

4 1.

198 Integrating Computers and Problem Posing in Mathematics Teacher Education

Fig. 7.8. Using Wolfram Alpha in developing formula (7.22).

Fig. 7.9. A spreadsheet generates a 5-cycle formed by the sequence LD

, ,

.

The spreadsheet of Fig. 7.9 (see Appendix for programming details) is designed to generate the last digits of the sequence , , for any pair (m, k). In particular, when m = 8, and k = 10, the spreadsheet generates the cycle {1, 5, 5, 1, 3} – a permutation with exactly three rises and two falls as in the chain of the inequalities 1 < 5 = 5 > 1 < 3 the signs “” occur exactly twice and once, respectively. Using this spreadsheet,

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one can pose a variety of problems about the m-gonal number sieves. Here are a few of them. Problem 7.5. The sum of the last digits of the first n terms of the triangular number sieve of order three is equal to 24. Find n. TITE scenario. One can begin with a TI part by developing the sequences , , , , , , , , and , 4 3 8 7 , then use a spreadsheet (Fig. 7.10) to model , the sequence , (column B), use formula (7.1) to generate the sequence LD , column D), and finally to carry out the summation of consecutive terms of the last sequence to get 24 (cell E9). The answer, n = 8, appears in cell A9.

Fig. 7.10. The TE part of solving Problem 7.5.

Problem 7.6. The sum of the last digits of the first 10 terms of the triangular number sieve of order k is equal to 30. Find the value of k. Does the problem have more than one correct answer? Why or why not? TITE scenario. A TI part begins with developing same formulas as in Problem 7.5 and then modeling within a spreadsheet (Fig. 7.11) the first ten terms of the triangular number sieves of orders two and three, their last digits, and the corresponding sums to see that the sums assume the

200 Integrating Computers and Problem Posing in Mathematics Teacher Education

value of 30 both for k = 2 and k = 3, thereby indicating that the problem has more than one correct answer.

Fig. 7.11. The TE part of solving Problem 7.6.

7.4 Patterns in the behavior of the greatest common divisors of two polygonal numbers Pairs of certain polygonal number sieves can be explored using their greatest common divisors (GCDs) as a characteristic of the corresponding sieve. As was already mentioned in Chapter 3, section 3.4, the spreadsheet provides a simple medium for such explorations which, in turn, can motivate seeking a formal explanation of a pattern observed. For example, the triangular number sieve of order one can be characterized by the cycle {1, 3, 1} formed by the GCDs of two consecutive terms of the sieve. As shown in the spreadsheet of Fig. 7.12, the sequence GCD , , forms the cycle {1, 3, 1} which differs , from the cycle {1, 1, 3} formed by the sequence GCD , , as , shown in the spreadsheet of Fig. 3.8 (Chapter 3). At the same time, the terms of the sequence GCD , , do not form any easy-to, recognize pattern. Likewise, the pairs of square numbers separated by another square number are either relatively prime or have the number 4 as their GCD. Indeed, GCD(1, 9) = GCD(9, 25) = 1 and GCD(4, 16) = GCD(16, 36) = 4. As shown in the spreadsheet of Fig. 7.13, the sequence GCD , forms the cycle {1, 4}. The above observations lead to posing the following TITE problems.

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Problem 7.7. Find the GCDs between two terms of the triangular number sieve separated by another term of the sieve, that is, GCD , , , . Use a spreadsheet to model the sequence GCD , , . Explain why the , . Show that sequence forms the cycle {1, 3, 1}. Problem 7.8. Find the GCDs between two square numbers separated by another square number, that is, GCD , . Use a spreadsheet to GCD , . Show that . Explain model the sequence why the sequence forms the cycle {1, 4}.

Fig. 7.12. GCD , , , forms the cycle {1,3, 1}.

Fig. 7.13. GCD , forms the cycle {1, 4}.

Similarly, one can use a spreadsheet to pose a problem involving GCDs of the terms of the pentagonal number sieves. Through this exploration, the following problems can be formulated.

202 Integrating Computers and Problem Posing in Mathematics Teacher Education

Problem 7.9. Find the GCDs between two pentagonal numbers separated by another pentagonal number, that is, GCD , . Use a spreadsheet GCD , . Show using a spreadsheet to model the sequence and formal reasoning that 35 for all n = 1, 2, 3, … . Problem 7.10. Find the GCDs between two consecutive terms of the pentagonal number sieve of order one, that is, GCD , , , . Use a spreadsheet to model the sequence GCD , , , , . Show through the joint use of a spreadsheet and formal reasoning that 35 for all n = 1, 2, 3, … . , 7.5 Exploring sequences formed by the sums of powers of integers This chapter started with considering the sum of the first 2006 fifth powers of integers. The sum was explored numerically. Likewise, similar sums can be considered and closed formulas for such sums can be generated by computer algebra software to be explored symbolically in the context of TITE problem posing and solving (e.g., see Chapter 3, section 3.9). Is there a context through which such sums can be introduced? What kind of counting problems can lead to the summation of the powers of integers? In the previous sections, 5-digit strings have been considered as cycles formed by the last digits of the terms of various numeric sequences. These cycles turned out to be permutations of digits in a string possessing special combinatorial properties formulated in terms of rises and falls induced in the elements of a cycle. In most of the cases that were considered, the cycles were permutations of five elements, only three of which were different. A question to be explored in this section can be formulated as follows: If there are five different digits available, what is the total number of different 5-digit strings that can be constructed out of those digits? To answer this question, five cases need to be considered. Case 1. A single digit is available to construct a 5-digit string. In this case, there exists a single string only.

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Case 2. Two different digits are available to construct a 5-digit string. In this case, each element of a string can be selected in two ways, so that by the Rule of Product (Chapter 1, section 1.3) there exist 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 2 strings of length five. Case 3. Three different digits are available to construct a 5-digit string. In this case, each element of a string can be selected in three ways, so that by the Rule of Product there exist 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 3 strings of length five. Case 4. Four different digits are available to construct a 5-digit string. In this case, each element of a string can be selected in four ways, so that by the Rule of Product there exist 4 ∙ 4 ∙ 4 ∙ 4 ∙ 4 4 strings of length five. Case 5. Five different digits are available to construct a 5-digit string. In this case, each element of a string can be selected in five ways, so that by the Rule of Product there exist 5 ∙ 5 ∙ 5 ∙ 5 ∙ 5 5 strings of length five. In all, there exist 1 2 3 4 5 4425 (the fifth number in the table of Fig. 7.2) different 5-digit strings that can be constructed using five different digits in all possible ways. In general terms, the problem of constructing strings of length r out of n different symbols can be formulated as follows: Find the number of all possible r-samples with repetitions of elements of different types varying in the range 1 through n. As was shown in the case of the strings of length five, the sum 1

2

3

⋯

(7.23)

provides the total number of such samples. In particular, when n = 2006 and r = 5 we have the sum mentioned at the beginning of section 7.1. Historically, the summation of powers of integers represented an advanced problem-solving context, with lower sums (r ≤ 4) already known by the 15th century [Edwards, 1982b]. The development of combinatorial and recursive techniques in the 17th and 18th centuries allowed for the general case formulas [Kotiah, 1993]. While there exist many ways of using technology in developing formulas for the sums for small values of r using empirical induction (e.g., by using spreadsheet

204 Integrating Computers and Problem Posing in Mathematics Teacher Education

modeling as shown in [Abramovich, 2011]), in the presence of powerful digital tools like Wolfram Alpha, capable of sophisticated symbolic computations, when the formulas can be ‘discovered’ almost at the push of a button, the summation activities using sums (7.23) for different values of r have to be revisited under the umbrella of TITE problemposing. Some activities of that kind were already discussed in Chapter 3, section 3.9. 7.6 Exploring sieves developed from the sums of powers of integers One may wonder whether any interesting pattern in the behavior of the last digits could be observed in the context of exploring the sequences formed by the sums of type (7.23) for different values of r as n varies within a large range. For example, when r = 1 we have the case of the triangular numbers the sieves of which were explored in sections 7.3.1 ∑ are and 7.3.2. Let r = 2. Then the terms of the sequence defined by the formula (7.24) which generates the numbers 1, 5, 14, 30, 55, 91, 140, …. Eliminating every second term from (7.24) yields the sequence 1, 14, 55, 140, … the general term of which can be found by substituting 2n – 1 for n in (7.24) yielding ,

.

(7.25)

Formula (7.25) represents the sieve of order one developed from the sequence and it generates the terms of the sieve in the far-left part of the spreadsheet of Fig. 7.14. Immediately to the right, the sequence LD , has been generated through formula (7.1) revealing the cycle {1, 4, 5, 0, 5, 6, 9, 0, 5, 0} of length ten. Note, that the OEIS® includes the sequence of numbers 1, 14, 55, 140, … (called structured rhombic dodecahedral numbers) and provides a closed formula for its general term identical to (7.25).

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The next step of the explorations is to develop formula for the sequence , . This can be done in two ways. The first way, which may be considered a TI part of the activity, is to substitute 2n – 1 for n in the expression for , . Another way is to enter into the input box of Wolfram Alpha the command “1, 55, 285, 819, 1785, …” to get

,

.

(7.26)

Formula (7.26) was used in the spreadsheet of Fig. 7.14 to generate consecutive terms of the sieve of order two developed from the sequence (the third from the left column of numbers). In turn, the sequence LD , located immediately to the right forms the cycle {1, 5, 5, 9, 5} of length five. Note, that the sequence 1, 55, 285, 819, 1785, … is not included into the OEIS® (higher order sieves developed from are not included as well). Now, using Wolfram Alpha one can get the formula

,

(7.27)

for the sieve of order three developed from the sequence . Formula (7.27) was used by the spreadsheet of Fig. 7.14 to model this sieve numerically (the fifth from the left column of numbers: 1, 285, 1785, 5525, 12529, …). Immediately to the right (the sixth from the left column of numbers) one can see that the sequence LD , forms the cycle {1, 5, 5, 5, 9} of length five. The spreadsheet of Fig. 7.14 shows one more step that defined the sieve of order four developed from the sequence . Continuing in the same vein, one can get the formula ,



(7.28)

for the sieve of order four used to generate numbers in the far-right part of Fig. 7.14. In turn, the sequence LD , forms the cycle {1, 5, 9, 5, 5} of length five.

206 Integrating Computers and Problem Posing in Mathematics Teacher Education

.

Fig. 7.14. Exploring the sieves developed from the sequence

Note that each of the tree cycles – {1, 5, 5, 9, 5}, {1, 5, 5, 5, 9} and {1, 5, 9, 5, 5} – are permutations with three rises as the sign “H$2,A4-1," ")) – copied down to cell A10. (C4) → =SUM(B4:B19); (D4) → =1-C4; (G2) → =2*A2-(B2+C2+1); (H2) → =A2-C2; (I2) → =A2-B2; (J3) → =J2*C4; (J4) → =J2*D4; (K3) → = J2*C2/(B2+C2); (K4) → =J2*B2/(B2+C2).

209

210 Integrating Computers and Problem Posing in Mathematics Teacher Education

Spreadsheet programming for Figures 1.7 and 1.8. (A1) → =COUNT(C3:H8). Cell B2 is slider-controlled and given name n. (D2) → =IF(AND(C2