Problem Posing and Solving for Mathematically Gifted and Interested Students: Best Practices, Research and Enrichment 3658410604, 9783658410605

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Deniz Sarikaya · Lukas Baumanns · Karl Heuer · Benjamin Rott   Editors

Problem Posing and Solving for Mathematically Gifted and Interested Students Best Practices, Research and Enrichment

Problem Posing and Solving for Mathematically Gifted and Interested Students

Deniz Sarikaya · Lukas Baumanns · Karl Heuer · Benjamin Rott Editors

Problem Posing and Solving for Mathematically Gifted and Interested Students Best Practices, Research and Enrichment

Editors Deniz Sarikaya Centre for Logic & Philosophy of Science Vrije Universiteit Brussel Brussels, Belgium Karl Heuer Department of Applied Mathematics and Computer Science Technical University of Denmark Kongens Lyngby, Denmark

Lukas Baumanns Institute of Mathematics Education University of Cologne Cologne, Germany Benjamin Rott Institute of Mathematics Education University of Cologne Cologne, Germany

ISBN 978-3-658-41060-5 ISBN 978-3-658-41061-2 (eBook) https://doi.org/10.1007/978-3-658-41061-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Spektrum imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Foreword

Mathematical problem posing, that is the generation of new and reformulation of given tasks that can be solved by mathematical means (Silver, 1994), is a genuine mathematical activity. It has been emphasized as an important mathematical activity by mathematicians (Cantor, 1867; Halmos, 1980) as well as mathematics educators (Kilpatrick, 1987; Brown & Walter, 2005). As an important companion to problem solving (Pólya, 1945), it can encourage flexible thinking, improve problem-solving skills, and sharpen learners’ understanding of mathematical contents (English, 1997). Therefore, this activity, which is a central skill of mathematicians, is often associated with mathematical creativity and giftedness. On January 22nd and 23rd, 2021, the editors of this book organized an International Symposium entitled Problem Posing for Mathematically Gifted Children, which was held online. The goals of this symposium were to exchange ideas, experiences, and research results between mathematicians, mathematics educators, and practitioners working on problem posing in the context of mathematical giftedness, and to establish scientific exchange between those groups. Specifically, ideas from the field for the enrichment of mathematically gifted children as well as findings from research on problem posing of or for mathematically gifted children were highlighted. With this aim, nine mathematicians and mathematics educators from five different countries came together over two days to present results and exchange ideas on best practices, research, and enrichment programs on problem posing and solving for mathematically gifted and interested students. The program of the symposium was as follows:

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vi

Foreword

Time (UTC+1)

Friday, January 22, 2021

Saturday, January 23, 2021

15:40–16:00

Opening: Incl. greetings by the President of the MCG: Marianne Nolte

16:00–18:00

Panel on the Mathematics Olympiad: Konrad Engel, Klaus Henning, Uwe Leck

Panel: How to work with students in times of corona: Torsten Fritzlar & Marianne Nolte

18:00–18:30

Break

Roza Leikin

18:30–19:10

Boris Koichu

Break

19:20–20:00

Terese Marianne Olga Nielsen

Igor’ Kontorovich

20:00–21:00

Dinner + Open Discussion

Dinner + Discussion of future activities

Recordings have been made and slides uploaded from numerous presentations at the symposium, which can be viewed and downloaded from the following official symposium website: problem-posing.weebly.com. Links to the video recordings of most presentations and can be found on that homepage. This book is much more than simple conference proceedings. It features articles from authors who were not present at the conference and not every talk developed into an article. To ensure the quality and of the contributions and scientific rigor, all articles were reviewed (single blinded) by at least two experienced scientist; at least one of those was a scholar who is not an author in this book. All articles within each part are ordered alphabetically. In the following, we will summarize in condensed form the contents of the chapters in the two parts of this book. Part I: “Perspectives from Research” might be seen as the more theoretical foundation, consisting of empirical work from math educators. This does not mean, that these articles do not engage with practice. Instead, there are both studies of concrete enrichment programs and more general considerations. Lukas Baumanns examines problem posing from a subject matter didactics perspective. In his paper, he first identifies four problem-posing activities based on a literature review. Subsequently, these four activities are illustrated by mathematical examples, called miniatures. Finally, the state of research on these four activities is summarized. Jason Cooper, Boris Koichu, Mirela Widder, Sarel Aiber, Yonah Amir, Aamer Badarneh, Menucha Farber, Michael Gorodin, Orly Gottlib, Esther

Foreword

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Gruenhut, and Fatmeh Hihi focus on a collaborative problem-posing model. This model aims at the development of a model that makes posed problems attractive and valuable in the eyes of teachers, teacher educators, and mathematicians. They describe principles that can contribute to the success of this challenging endeavor of problem posing in a diverse group of posers. Igor’ Kontorovich focuses in his conceptual chapter on practices of experienced problem posers for mathematics competitions. This perspective is used to encourage mathematics education research to investigate this field to gain insights into problem posing in the educational context. Ahlam Mahagna, Abraham Berman, and Roza Leikin present the results of a study dealing with the impact of an enrichment course on Grade 9 students’ problem-solving skills, creativity, and attitudes towards learning mathematics. They found a positive impact of the enrichment course on all three constructs studied. Marianne Nolte provides an overview of questions on diagnostics and procedures of high mathematical talent—particularly in times of COVID-19 pandemic. Her paper concludes to favor multidimensional approaches with a focus on special mathematical tests which tests mathematical creativity, the recognition of patterns and structures and the capability to generalize and explain ideas as well as an intelligence test. Katrin Vorhölter and Malte Pamperien present findings on the activity of problem posing in the context of PriSMA math circles—a concept for identifying and supporting mathematically gifted children. They found that students not only posed problems that resulted from a given problem-posing situation, but also that new problems resulted independently of the problem-posing situation through personal relations to the situation presented in the task. Part II: “Perspectives from Enrichment Programs: Best Practices” contains articles that report experiences from specific enrichment programs. Here a greater focus lies on the actual material and background information on the activities. Eszter Bóra and Péter Juhász offer a window into the world of mathematical enrichment programs in Hungary and their method of problem threads. The participants work on multiple threads in one session. We learn about the camps, concrete question and get a greater embedding into the Hungarian heuristic tradition. Terese M. O. Nielsen introduces both the Georg Mohr Contest and the Science Talenter Camps. We learn that the problems of the contest can well be used in the camps. We see specific problems and how students engage with them. The next three articles all offer three case studies of concrete material. Those are: Merlin Carl and Michael Schmitz: Discoveries in a 10-adic number world.

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Foreword

Merlin Carl, Uwe Leck, Hinrich Lorenzen, and Michael Schmitz: Combinatorial Checkerboard Problems for Kids Merlin Carl, Uwe Leck, Hinrich Lorenzen, and Michael Schmitz: Elementary Length Formulas for Triangles and Quadrilaterals. The materials are used in a plurality of different enrichment formats, including the Schülerakademie at the University of Flensburg, Workshops in the US, Japan, and Wales. We would like to thank all persons who presented at the symposium and enabled a valuable exchange of ideas in the first place. We would also like to thank all authors of this book, without whom such an edited volume would not have been possible. Finally, special acknowledgment goes to the numerous anonymous and external reviewers who have contributed significantly to the quality assurance of the present contributions. Deniz Sarikaya’s work was supported by the FWO Grant: FWOAL950. Brussels, Belgium Cologne, Germany Lyngby, Denmark Cologne, Germany December 2022

Deniz Sarikaya Lukas Baumanns Karl Heuer Benjamin Rott

Contents

Perspectives from Research Four Mathematical Miniatures on Problem Posing . . . . . . . . . . . . . . . . . . Lukas Baumanns Many Chefs in the Kitchen—a Collaborative Model for Problem-Posing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jason Cooper, Boris Koichu, Mirela Widder, Sarel Aiber, Yonah Amir, Aamer Badarneh, Menucha Farber, Michael Gorodin, Orly Gottlib, Esther Gruenhut, and Fatmeh Hihi Would Specialist Problem Posers Endorse Problem-Posing Situations that We Design for Learners? Does It Matter? . . . . . . . . . . . . . Igor’ Kontorovich The Impact of an Enrichment Course in Mathematics on Students’ Problem-Solving Skills, Creativity, and Attitudes Towards Learning Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ahlam Mahagna, Abraham Berman, and Roza Leikin Questions About Fostering and Identifying of Mathematically Promising Students in Times of Covid-19 Pandemic . . . . . . . . . . . . . . . . . Marianne Nolte Problem Posing as an Integral Part for the Support of Mathematically Highly Gifted Teenagers Within the PriSMa Math Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Katrin Vorhölter and Malte Pamperien

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Perspectives from Enrichment Programs: Best Practices How to Use Motion as a Problem-Solving Tool? Problems from the Pósa Camps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eszter Bóra and Péter Juhász

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Using Maths Competition Problems at Camps . . . . . . . . . . . . . . . . . . . . . . Terese M. O. Nielsen

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Teaching Units for Mathematical Enrichment Activities . . . . . . . . . . . . . . Merlin Carl, Uwe Leck, Hinrich Lorenzen, and Michael Schmitz

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Contributors

Sarel Aiber Science Teaching, Weizmann Institute of Science, Rehovot, Israel Yonah Amir Science Teaching, Weizmann Institute of Science, Rehovot, Israel Aamer Badarneh Science Teaching, Weizmann Institute of Science, Rehovot, Israel Lukas Baumanns Faculty of Mathematics and Natural Sciences, Institute of Mathematics Education, University of Cologne, Cologne, Germany Abraham Berman Mathematics, Technion- Israel Institute of Technology, Haifa, Israel Eszter Bóra MTA-Renyi-ELTE Research Group in Mathematics Education, Alfréd Rényi Institute of Mathematics, Budapest, Hungary Merlin Carl Institut für Mathematik, Europa-Universität Flensburg, Flensburg, Germany Jason Cooper Science Teaching, Weizmann Institute of Science, Rehovot, Israel Menucha Farber Science Teaching, Weizmann Institute of Science, Rehovot, Israel Michael Gorodin Science Teaching, Weizmann Institute of Science, Rehovot, Israel Orly Gottlib Science Teaching, Weizmann Institute of Science, Rehovot, Israel Esther Gruenhut Science Teaching, Weizmann Institute of Science, Rehovot, Israel

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Contributors

Fatmeh Hihi Science Teaching, Weizmann Institute of Science, Rehovot, Israel Péter Juhász MTA-Renyi-ELTE Research Group in Mathematics Education, Alfréd Rényi Institute of Mathematics, Budapest, Hungary Boris Koichu Science Teaching, Weizmann Institute of Science, Rehovot, Israel Igor’ Kontorovich Department of Mathematics, The University of Auckland, Auckland, New Zealand Uwe Leck Institut für Mathematik, Europa-Universität Flensburg, Flensburg, Germany Roza Leikin Faculty of Education, University of Haifa, Haifa, Israel Hinrich Lorenzen Institut für Mathematik, Europa-Universität Flensburg, Flensburg, Germany Ahlam Mahagna Mathematics Education, University of Haifa, Haifa, Israel Terese M. O. Nielsen Science Talenter, Astra, Sorø, Denmark Marianne Nolte Faculty of Education, University of Hamburg, Hamburg, Deutschland Malte Pamperien Fakultät für Erziehungswissenschaft, Universität Hamburg, Hamburg, Germany Michael Schmitz Institut für Mathematik, Europa-Universität Flensburg, Flensburg, Germany Katrin Vorhölter Fakultät für Erziehungswissenschaft, Universität Hamburg, Hamburg, Germany Mirela Widder Science Teaching, Weizmann Institute of Science, Rehovot, Israel

Perspectives from Research

Four Mathematical Miniatures on Problem Posing Lukas Baumanns

1

Introduction

Posing questions has always been a central part of constructing new knowledge for scientific progress but also the individual learning process. Already Einstein and Infeld (1938) state: “Galileo formulated the problem of determining the velocity of light, but did not solve it. The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advance in science.” (p. 95)

Also in mathematics, posing problems in itself represents a central value. Cantor (1867) already stated as the third thesis of his dissertation: “In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi.” (p. 26). This translates as: In mathematics, the art of posing a question is of greater value than solving it. We find similar statements from Lang (1989, p. 70), Poincaré (1973, p. 246), or Hilbert (1900, p. 262) who emphasize the importance of posing problems in the field of mathematics. For example, Erd˝os is considered to be one of the most prolific L. Baumanns (B) Faculty of Mathematics and Natural Sciences, Institute of Mathematics Education, University of Cologne, Cologne, Germany E-mail: [email protected] © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 D. Sarikaya et al. (eds.), Problem Posing and Solving for Mathematically Gifted and Interested Students, https://doi.org/10.1007/978-3-658-41061-2_1

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mathematicians who, in addition to solving many problems, was also, and perhaps especially, known for posing problems in various fields (Ramanujam, 2013). In mathematics education research, posing problems has been mentioned at least since dealing with problem solving (Pólya, 1957; Schoenfeld, 1985, 1992). Pioneering conceptual considerations of problem posing can be found in Silver (1994), Stoyanova and Ellerton (1996), or Brown and Walter (1983). For instance, Silver (1994) says, problem posing is the generation of new as well as the reformulation of given problems and occurs before, during, or after problem solving. Looking at current empirical research in problem posing (Baumanns & Rott, 2022; Lee, 2021), a dilution of the original idea of problem posing is partly noticeable. As Silver (2013) writes 19 years after his seminal article: “Is the time ripe for the field to make sharper distinctions among the several manifestations of mathematical problem posing as a phenomenon? To what extent is the activity of problem posing that a teacher does in order to provide an appropriate task to her students similar to or different from the activity of problem posing that a teacher does when posing a problem for herself, or the activity of problem posing when her students pose problems for their classmates? And what does any of this kind of problem posing have to do with what happens when a person reformulates a problem during the course of solving a complex problem or when one looks back at a problem solution and engages in creative reformulation (a la Polya or Brown and Walter)?”

This paper is an attempt to look at Silver’s (2013) and other different problem-posing activities through mathematical miniatures in order to gain a subject-matter didactic perspective on problem posing through reflection on the mathematical activity itself. Therefore, this paper has the following aims: 1. To identify problem-posing activities based on definitions and concepts from mathematics education. 2. To present these identified problem-posing activities through selected mathematical miniatures. 3. To summarize the results of empirical research within the identified problemposing activities. Figure 1 summarizes the structure of this paper: First, key definitions and conceptions of problem posing from mathematics education are summarized. From these definitions and conceptions, four different problem-posing activities are elaborated based on the different goals problem posers pursue: (1) Problem posing as generating new problems, (2) problem posing as reformulating a given problem for problem solving, (3) problem posing as reformulating a given problem for investigation, and

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Fig. 1 Structure of this paper

(4) problem posing as constructing tasks for others. Each of these problem-posing activities is first illustrated with a mathematical miniature. Findings of empirical studies dealing with the respective problem-posing activity are then summarized. The paper ends with a conclusion.

2

Problem Posing in Mathematics Education—Definitions and Conceptions

We find the first considerations on problem posing in mathematics education in the seminal works on problem solving. Problem posing in this context is seen as a partial aspect of problem solving. In his short dictionary of heuristics, Pólya (1957) describes at least two functions of problem posing. On the one hand, he describes problem posing for problem solving by varying the problem (p. 209). Pólya gives several examples of how varying a problem can help solve it. Specifically, as one example in this section of his seminal book How to solve it, he solved the problem of constructing a trapezoid using only the four given sides by posing two auxiliary problems through variation, making the trapezoid first a triangle and finally a parallelogram. On the other hand, Pólya asks the question “Can you use the result, or the method, for some other problem?” (p. 64). With this, he addresses that one should reap the fruits after solving a problem. In this context, he describes three paths for new problems in terms of their accessibility and interestingness. (1) Using the canonical variation strategies generalization, specialization, analogy, decomposing, and recombining, learners can get to new problems. However, these problems are

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rarely accessible as they soon get too difficult. (2) It is also possible to pose new problems, which can be solved using the same method that was used to solve the initial problem. However, according to Pólya, these are rarely interesting as there are no structural new insights. (3) What is difficult is finding new problems that are accessible but remain interesting. Pólya provides a few examples where this endeavor has been successful. Schoenfeld takes up Pólya’s ideas in his seminal remarks on problem solving (1985, 1989, 1992). As he states, problem posing helps for problem solving, identical to what we have already seen in Pólya. He suggests, in the context of problem solving, that if difficulties arise, one should use modification to arrive at simpler problems (1985, p. 102). The focused investigation of problem posing—and thus the terminological definition of this concept of activity—began in the 1980s (Walter & Brown, 1977; Butts, 1980; Brown & Walter, 1983; Ellerton, 1986; Kilpatrick, 1987). Butts (1980) pointed out that the way problems are posed significantly affects the problem solver’s motivation to solve the problem, as well as his or her comprehension of the key concepts underlying the problem. Of particular note is certainly Brown & Walter’s well-known “What-If-Not” strategy (Brown & Walter, 1983, 2005). This strategy involves going through five stages: 1) Choosing a starting point, 2) listing attributes, 3) What-If-Not-ing, 4) question asking or problem posing, and 5) analyzing the problem.1 In Germany, Schupp (2002) takes up the idea by Brown & Walter. He proposes 24 task variation strategies by means of which students can pursue the activity of task variation in a regulated way. Among these strategies we find analogize, generalize, specialize, but also something like sensemaking, reverse, or iterate. In subsequent years, researchers in mathematics education have attempted to bring order and structure to this newly conceptualized field. Over the years, different definitions of problem posing have emerged. Table 1 summarizes some of these widely cited definitions. These definitions always refer to activities of posing problems in the field of mathematics. Problem refers in this context to all tasks on the spectrum between routine task and non-routine problem. By non-routine problems, in contrast to routine tasks, we mean tasks for which no procedure for a solution is known and strategies for a solution have to be found and applied (Schoenfeld, 1985). The definition of Silver (1994) includes two different activities: Generating and reformulating. According to Silver (1994), problem posing can occur before, during, or after problem solving. Problem posing occurs before problem solving when the goal is not the solution of a problem but the creation of a new problem. This refers to the generation of a new problem. Problem posing occurs during problem solving

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when someone is stuck in a problem-solving process and tries to make that problem more accessible by reformulating it. Problem posing can also occur after problem solving. This refers to the activity of investigating a given problem to get more generalized insights into it. Both problem posing during and after problem solving refers to the reformulation of a given problem. Stoyanova and Ellerton (1996) provide a more student-oriented definition (see Table 1). This definition is based on individual mathematical experiences and sees problem posing as a process of interpreting concrete situations. A situation is a not well-structured problem in the sense that the goal cannot be determined by all given elements and relationships (Stoyanova, 1997, p. 5). Stoyanova and Ellerton (1996) distinguish between free, semi-structured, and structured problem-posing

Table 1 Problem-posing definitions Source

Conception

Silver (1994)

“Problem posing refers to both the generation of new problems and the re-formulation, of given problems.” “[M]athematical problem posing will be defined as the process by which, on the basis of mathematical experience, students construct personal interpretations of concrete situations and formulate them as meaningful mathematical problems” “[P]roblem posing means an accomplishment that consists of constructing a problem that satisfies the following three conditions: (a) it somehow differs from the problems that appear in the resources available to the teacher; (b) it has not been approached by the students; and (c) it can be used in order to fulfill teaching needs that otherwise could be difficult to fulfill.” “By problem posing in mathematics education, we refer to several related types of activity that entail or support teachers and students formulating (or reformulating) and expressing a problem or task based on a particular context (which we refer to as the problem context or problem situation).”

Stoyanova and Ellerton (1996)

Klinshtern et al. (2015)

Cai and Hwang (2020)

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situations, depending on their degree of given information. Free situations provoke the activity of posing problems out of a given, naturalistic, or constructed situation without any restrictions. In semi-structured situations, posers are invited to explore the structure of an open situation by using mathematical knowledge, skills, and concepts of previous mathematical experiences. In structured situations, people are asked to pose further problems based on a specific problem, for example by varying its conditions. Referring to the conceptualization by Koichu (2020), the problem-posing situations can be seen as didactical, i.e. problem posing is a goal by itself and is explicitly demanded. Besides that, problem posing can be evoked a-didactically as an implicit goal. This applies, for example, for open-ended or inquiry-based problem-solving environments (Cifarelli & Cai, 2005; da Ponte & Henriques, 2013). Klinshtern et al. (2015) provide a more teacher-oriented definition. Their definition is the result of a study on what teachers mean by posing problems. For this reason, this definition also includes something like an educational goal of the posed problem. Cai and Hwang (2020) integrate student and teacher perspectives with their definition. Their conceptualization also includes, in broad terms, the distinction between generating and reformulating in Silver’s (1994) definition, as well as the openness of the problem situation inherent in Stoyanova and Ellerton’s (1996) definition. In the following, we want to cover different perspectives on the activity of problem posing. In this context, we want to use the goals pursued by the respective problem-posing activity as a criterion for differentiation. General educational goals related to the activity of problem posing are, among other things, to give an authentic idea of what doing mathematics is and to give students a positive self-concept (Schupp, 2002). In this context, however, by goals we do not mean overarching educational goals, but rather focused goals of individual problem-posing sequences. In the following, we will go into more detail about what is meant by this. To specify what goals we are talking about, we make use of Silver’s (1994) conceptualization described earlier, which says problem posing can take place as the generation of new and reformulation of given problems before, during, and after problem solving. In problem posing before problem solving, the goal is to create a new problem. During problem solving, problem posing aims at making a problem more accessible for a solution. In problem posing after problem solving, the goal is to get a more general insight into a mathematical field or to expand the scope of the mathematical content. We want to bring another education-related perspective to problem posing. We already saw this perspective in the conceptions by Klinshtern et al. (2015) and Cai and Hwang (2020). Problem posing can be done not only for oneself, but also for others, that is teachers pose problems for pupils, lecturers for

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students, or expert problem posers for the contestants of mathematics competitions. This kind of problem posing aims at achieving an educational goal. The goal is to stimulate the construction of new knowledge concerning a mathematical content through appropriately posed problems. These structural considerations gave rise to the following list of different perspectives on problem posing, which is discussed in the following sections: 1. 2. 3. 4.

Problem posing as generating new problems Problem posing as reformulating a given problem for problem solving Problem posing as reformulating a given problem for investigation Problem posing as constructing tasks for others

In each section, the corresponding problem-posing activity will first be shown as an example in the form of a mathematical miniature. Then it will be located more generally in the research field by presenting studies that investigate the specific problem-posing activity. The four presented perspectives are by no means separable from one another. We will see that there are areas of intersection of different sizes where the perspectives overlap.

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Four Mathematical Miniatures

3.1

Problem Posing as Generating New Problems

Mathematical Miniature To describe the free activity of generating new problems as problem posing, we take a situation where no initial problem is given (see Brown & Walter, 2005). Geoboard

Pose problems for the following .5 × 5 geoboard.

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Numerous questions can now be asked about this geoboard. We would like to develop a few ideas in the following. We do not want to pursue a detailed solution to the problems in every case, as the focus is rather on the process of posing. Let us assume that the horizontal and vertical distance of the points has the dimensionless length 1. If you now connect different points on √ the geoboard, you will find different lengths. Of course all lengths from 1 to 4, . 2 for a diagonal connection and so on. This does suggest the question of how many different lengths you can already find on the geoboard. If you connect any two points on the geoboard by a straight line, you will find different lengths. How many different lengths can you find?

Now that we’ve dealt with different lengths on the geoboard, another obvious question arises because of the square shape of the board. We find different squares on √ √ the geoboard. Of course a .1 × 1 square, the big .4 × 4 square, or a . 2 × 2 square. But what about other, integer and non-integer side lengths? What is the total number of squares that you can find when connecting dots on a .5 × 5 geoboard? We first look at the squares whose sides are parallel to the edge of the geoboard. We find one .4 × 4 square, four .3 × 3 squares, nine .2 × 2 square, and sixteen .1 × 1 square. These quantities are √ obviously the square numbers. In√addition, we find one square .2 2, two squares with side length . 10, eight squares with side with side length √ √ length . 5, and nine squares with side length . 2. So, in total, there are 50 different squares on a .5 × 5 geoboard. These squares are shown graphically in the following:

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So how can we be sure that we have found all the squares? And if we could answer that, could we generalize these thoughts? Can we identify structures that allow us to determine the number of different squares on a .n × n geoboard? But with such considerations, we are at Problem posing as reformulating a given problem for investigation which we want to consider in more detail later. Now we have dealt more intensively with squares, but what about triangles? Right-angled or isosceles triangles can be found quickly. An equilateral triangle is not so easy to find. So this problem could be formulated: Are there equilateral triangles on a .5 × 5 geoboard? Now we have posed numerous problems where the dots have been connected. One can quickly find lines within the .5 × 5 geoboard that do not meet any point. What does this look like when you extend the geoboard to a .n × n geoboard or even a .∞ × ∞ geoboard? In this context, there are at least two problems that could be investigated: Can I find a line on an unbounded .∞ × ∞ geoboard that does not intersect any of the points? Can I find a line that intersects only a finite number of points?

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Summarizing Empirical Research According to (Stoyanova & Ellerton, 1996), the given situation of the geoboard without a given initial task can be called semi-structured. Some studies use semistructured problem-posing situations or focus the activity of problem posing as generating new problems. For example, Cai and Hwang (2002) asked participants to pose an easy, medium, and hard problem on a specific dot pattern. They investigated the problem-posing performance of Chinese and US sixth-grade students and compared it to their problem-solving performance. They found a stronger link between problem-posing and problem-solving performance in the Chinese students than in US students. Bonotto (2013, see also Bonotto & Santo, 2015) chose a contextualized input. She investigates the potential of so-called artifacts (i.e. real-life objects like restaurant menus, advertisements, or TV guides) to stimulate critical and creative thinking. Bonotto (2013) investigated the relationship between problem-posing and problem-solving activities in 18 primary students and also assessed the use of these artifacts as a semi-structured situation. She concludes that these artifacts from the real world are an appropriate stimulus for problem-posing activities. However, the complexity of the specific artifact, its constraints, and variables should be analyzed before implementing it into the classroom. Silver et al. (1996) had participants explore a billiard context in their study. This context has also been frequently used in subsequent studies (Kontorovich et al., 2012; Koichu & Kontorovich, 2013). In this context, (Koichu & Kontorovich, 2013), for example, found that in two successful problem-posing processes when the posers are asked to pose an interesting problem, they also have their own interest in mind. Problem posing then initially consists of searching for an interesting mathematical phenomenon within the given semi-structured situation.

3.2

Problem Posing as Reformulating a Given Problem for Problem Solving

Mathematical Miniature As we pointed out above, Pólya (1957) and Schoenfeld (1985), although they never used the term problem posing explicitly, wrote extensively about modifying a problem for solving. Both have in common that for solving a problem they propose to pose a simpler problem. At Pólya (1957), this is done in the Devising a plan phase. With (Schoenfeld, 1985), this takes place in the Exploration phase. Let’s look at an example problem for this.

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Example Problem

In how many paths can the words PROBLEM POSING be read in the figure below? P R R O O O B B B B L L L L L E E E E E E M M M M M M M P P P P P P O O O O O S S S S I I I N N G

So we are looking for the number of paths leading from the first P to the last G in this figure. The number of paths to read the words PROBLEM POSING does not seem easy to count. For this reason, we modify the task (Pólya, 1957; Schoenfeld, 1985) and solve a structurally similar, but a simpler task. Let us consider the word PROBL first. Consider an arbitrary letter in the figure. How many different paths originating at the top (P) lead to this letter?

We will now use the third letter L in the third row to answer this modified task. P R O

R O

B

O B

L To get from the P to on of the R’s in the second row of the figure, I obviously have exactly one option in each case. For the O’s that are on the very outside of this figure

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also remain only one option. The O in the middle, however, has two possibilities. From the P, I can go to the right or to the left, and then come to the middle. If I want to go to one of the B’s in the line below, I have three choices. Either I go over one of the outer O’s or I go one of the two paths over the middle O for I have two options. So slowly a pattern reveals itself. We modify the figure slightly by abstracting it. Instead of the letters, we fill the squares with the number of paths leading from the top to this point. 1 1 1

1 2

3

1 3

6 If we look at this figure, we find that any number other than 1 is the sum of the two adjacent numbers above it. But why is that? As an example, consider again the B in the figure above. If we want to reach B, starting from P, we have to pass through one of the two neighboring O’s above it. Once we have reached one of the two O’s, there is only one way to get to B from there. So: The total number of ways from P to B is the sum of ways to the two adjacent O’s above. Let us apply this rule to the initial problem: 1 1 1 1 1 1 1

3 4

7 28

6 35

1 4

10 20

15 21

1 3

10

5 6

1 2

1 6

15 35

1

5 21

1 7

56 70 56 28 84 126 126 84 210 252 210 462 462 924

Thus, there are 924 paths to read the words PROBLEM POSING in the respective figure. Of course, the interested reader should notice that this structure is a segment of Pascal’s triangle. In this exploration, we started with a problem that we could not solve at first. We have made it a structurally similar but simpler problem by modification. We were able to solve it by recognizing a pattern. This pattern could

Four Mathematical Miniatures on Problem Posing

15

be transferred to the initial problem because of the structural equality so that we could solve this problem as well. So we did problem posing as reformulating for problem solving.

Summarizing Empirical Research When we look at research on problem posing for problem solving, conceptual work stands out the most. The work of Pólya (1957) and Schoenfeld (1985) has already been referred to in this context. In fact, Duncker (1945, pp. 8–9) already pointed out that, for him, problem solving can be conceptualized as productive reformulating of a given initial problem. With regard to empirical research on problem posing as reformulating a given problem for problem solving, we see that in Xie and Masingila (2017) problem posing is quite explicitly intended to be used as a heurism for solving a problem. They state that “especially posing easier problems than the original one, enhances participants’ understanding of the structure of the given problem and further contributed to problem solving” (p. 108). Cifarelli and Sevim (2015) investigate how problem posing contributes to solvers’ problem-solving activity and conclude that problem posing should be emphasized by teachers as it is an integral part of problem solving.

3.3

Problem Posing as Reformulating a Given Problem for Investigation

Mathematical Miniature We want to illustrate this kind of problem posing with an example from number theory (cf. Ziegenbalg, 2014):

Sum of Consecutive Numbers

In how many different ways can you express 15 as the sum of consecutive numbers?

We first consider a possible solution to this task. We notice that we can express 15 as .7 + 8. Furthermore, let us first express 15 as a sum of equal summands. For this purpose it helps to look at the prime factorization or the divisors of 15. Besides 1, 15 is divisible by 3, 5, and 15. So, we can express 15 as .3 · 5 = 5 + 5 + 5, .5·3 = 3+3+3+3+3, and.15·1 = 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1.

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We can cleverly rearrange within these sums, so for example in the sum .5 + 5 + 5, we decrease the first summand by 1 and increase the last one by 1, so that the sum remains constant. This leads to .3 + 4 + 5 which is a sum of consecutive numbers. The following figure illustrates this process with figured numbers.

.→

For .3 + 3 + 3 + 3 + 3 we proceed analogously. The middle 3 remains, the second summand we decrease by 1, the fourth summand we increase by 1. The first summand we decrease by 2, the fifth summand we increase by 2. This leads to the sum .1 + 2 + 3 + 4 + 5. Also for the sum of 1’s, we proceed like that, leading to .(−6) + (−5) + (−4) + (−3) + (−2) + (−1) + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8. The negative integers cancel each other out and the sum .7 + 8 remains. This leads to: 15 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5 = 7 + 8 There do not seem to be any other solutions. Interestingly, there seems to be a relationship between the number of divisors and the number of ways to express a number as the sum of consecutive numbers. 15 has three divisors not equal to 1 and can be expressed in three ways by the sum of consecutive numbers. Let us now begin to get to a more general insight in the sense of investigating through reformulating this initial problem. For that, we pose a similar problem in which we consider a different number which we want to express as a sum of consecutive numbers with the same approach: In how many different ways can you express 20 as the sum of consecutive numbers? 20 is, besides 1, divisible by 2, 4, 5, 10, and 20. So, we can express 20 as .10 + 10, + 5 + 5 + 5, .4 + 4 + 4 + 4 + 4, .2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2, and .1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1. If we now want to make the same rearrangement as with the above sums of the number 15, we find that .5

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17

we do not succeed in every case..10+10 cannot be rearranged in such a way, since we have an even number of summands, with which we cannot rearrange with constant sum. The same applies for the sum.5+5+5+5,.2+2+2+2+2+2+2+2+2+2, and .1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1. Only in the case of .4 + 4 + 4 + 4 + 4 can we perform our procedure and arrive at the only solution of expressing 20 as the sum of consecutive numbers:.20 = 2+3+4+5+6. This means that not only the number of divisors plays a role, but also their parity. The number 15 has three odd divisors greater than 1, and consequently it can be written as a sum of consecutive numbers in three different ways. The number 20 has only one odd divisor above 1 and therefore one expression as the sum of consecutive numbers. If this pattern continues, there should be no expression as a sum of consecutive numbers for numbers that have only even divisors, for example, 4 .16 = 2 . So let’s pose the next problem: In how many different ways can you express 16 as the sum of consecutive numbers? 16 has, besides 1, the divisors 2, 4, 8, and 16 which leads to the sums.8+8,.4+4+4+4, .2+2+2+2+2+2+2+2, and.1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1.

According to our expectation, these sums cannot be cleverly rearrange, so that a sum of consecutive numbers results. Our assumption is now solidified here finally and we can state: Any natural number can be represented as a sum of consecutive numbers in as many ways as it has odd divisors .> 1. This is actually called Sylvester’s Theorem after the British mathematician James Joseph Sylvester.

Summarizing Empirical Research This adjacency of problem posing and problem solving suggested in this section is also explored in the context of an open problem situation by Cifarelli and Cai (2005). They observed two students solving a problem related to billiard and found that problem posing and problem solving were always intertwined. They state that “problem posing and solving appeared to evolve simultaneously, each informing and serving as a catalyst for the other as solution activity progressed” (p. 321). They made similar observations in a later study (Cifarelli & Sevim, 2015). Voica and Singer (2013) analyzed the products of 42 students with above-average mathematical abilities modifying a given initial problem. They found evidence of links between the quality of the students’ posed problems and their cognitive flexibility. Martinez-

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Luaces et al. (2019) also used a structured situation to encourage prospective teachers to enrich an initial problem by modifying it. They found out that the prospective teachers have been very creative in the reformulation of the given problem as well as the enrichment of the problem itself. However, some of the participants trivialized the initial problem. As Martinez-Luaces et al. (2019) states, this should not be criticized as these trivializations sometimes helped to get further insights into the problem itself.

3.4

Problem Posing as Constructing Tasks for Others

Mathematical Miniature In this section, we will look at what problem posing can look like when you do it for others rather than for yourself. As an example, we use a specific topic from number theory. A typical task in number theory is to determine all divisors of a number. Using the prime factorization of a natural number .n > 1, one can arrange the divisors of .n in a Hasse diagram representing the divisibility relation in the divisor set .Tn . Textbooks on elementary number theory often have exercises that involve Hasse diagrams (Padberg, 2008). In a Hasse diagram, there is a connection from .t1 ∈ Tn to .t2 ∈ Tn if and only if .t1 | t2 , t1 < t2 and there is no .s ∈ Tn with 4 .t1 | s and .s | t2 . In the following figure, the Hasse diagrams are given for .8 = 2 2 and .12 = 2 · 3. 12 8

4

2

1

6

4

3

2

1

As a hypothetical lecturer in a number theory course, I now want students to use the Hasse diagrams to represent the divisors of a number in a structured way. To understand the structure of Hasse diagrams, students should first draw divisor sets of certain numbers and their corresponding Hasse diagrams. To make the Hasse diagrams different from each other, I choose the task so that the number of prime factors of the numbers differs. This applies, for example, to the numbers .81 = 34 ,

Four Mathematical Miniatures on Problem Posing

19

= 23 · 13, and .350 = 2 · 52 · 7. Therefore, I first pose the following task for the students:

.104

Determine the divisor sets of the numbers 81, 104, and 350. Draw the corresponding Hasse diagrams.

This leads to the following solution: 104 81 350 8

52

27

70 26

9

14

4

50

10 35

3

13

25

2 7

2 1

175

1

5

1

It now occurs to us that for a number with four prime factors (e.g. .9828 = 22 · 33 · 7 · 13), we would have to draw a four-dimensional Hasse diagram. How could this look like? Students could come across this insight with the following task: Try to draw the Hasse diagram of the number 9828. Why is this difficult? Can you generalize these thoughts?

Now, to encourage a more flexible use of Hasse diagrams, I reverse the previous task on one side and generalize it1 :

1 Note that two strategies of task variation according to Schupp (2002) were used here, namely invert and generalize.

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L. Baumanns

Describe the set of all numbers whose Hasse diagrams have the following form:

From left to right, this leads to the following statements • All numbers .n ∈ N whose prime factorization is of the form .n = p 3 with any prime . p. • All numbers .n ∈ N whose prime factorization is of the form .n = p12 · p22 with any primes . p1 , p2 . • All numbers .n ∈ N whose prime factorization is of the form .n = p1 · p2 · p33 with any primes . p1 , p2 , p3 . From the Hasse diagrams we get the following relation: A natural number .n with the prime factorization .n = p1n 1 · p2n 2 · . . . · pkn k has .(n 1 + 1)(n 2 + 1) · . . . · (n k + 1) divisors. Accordingly, the number of divisors can be read directly from the prime factorization. This should also find its way into a posed task to the students. The Hasse diagram shows all divisors of a specific number. How does this help you to determine all divisors of 8128512? Because.8128512 = 211 ·34 ·72 it follows that 8128512 has.(11+1)·(4+1)·(2+1) = 180 divisors. These considerations show us that there is a relation between the number of divisors and the number of different prime factors of a number. This can also be used for a task, that could be posed at the end:

Four Mathematical Miniatures on Problem Posing

21

Show that there is no natural number with exactly 20 divisors and exactly 4 prime divisors.

To justify, let .k be a natural number with exactly 4 prime divisors, that is, a prime factorization of the form .k = p1n 1 · p2n 2 · p3n 3 · p4n 4 . Now if .k is to have exactly 20 divisors, .(n 1 + 1)(n 2 + 1)(n 3 + 1)(n 4 + 1) = 20 must be true. 20 has the prime factorization .20 = 22 · 5. Thus, the number 20 cannot be a product of four integers greater than one. This leads to a contradiction, which shows that there is no natural number with exactly 20 divisors.

Summarizing Empirical Research Looking at research done on problem posing as constructing tasks for others, we see several studies that investigate in- or pre-service teachers that are asked to pose problems for their students based on a specific given task or content. In this context, problem posing fulfills an educational goal because, for example, teachers want to achieve a certain learning goal with the students using tasks posed by the teacher (cf. Cruz, 2006) Problem posing can also be a diagnostic tool, because thoughtful posed problems can reveal students’ definitions and misconceptions (Chen et al., 2011; Tichá & Hošpesová, 2013). For example, Nicol and Crespo (2006) have found that two participants in their study expanded the mathematical content of the selected textbook tasks to make them more complex. The teachers were also asked to collect tasks for teaching based on the available textbooks. In this context, they indicated a preference for self-posed problems. Textbook authors also need to set numerous tasks of different kinds (e.g., exercises, contextualized tasks, proving tasks) on different topics. This constructing tasks for others also includes, of course, the creation of tasks for mathematics competitions (Poulos, 2017; Kontorovich, 2020; Kontorovich & Koichu, 2016; Sharygin, 2001). For example, Kontorovich and Koichu (2016) observed an expert problem poser posing problems for mathematics competition. They have found that the expert problem poser draws new problems from a family of familiar problems. Moreover, the expert problem poser was concerned that the solution is perceived as novel and surprising not only to potential solvers but also to himself.

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Conclusion

The present paper had three aims: 1) To identify problem-posing activities based on definitions and concepts from mathematics education, 2) to present these identified problem-posing activities through selected mathematical miniatures, and 3) to summarize the results of empirical research within the identified problem-posing activities. Based on seminal definitions and conception on problem posing in mathematics education (Silver, 1994; Stoyanova & Ellerton, 1996; Cai & Hwang, 2020; Brown & Walter, 1983; Klinshtern et al., 2015), four problem-posing activities were identified: 1. 2. 3. 4.

Problem posing as generating new problems Problem posing as reformulating a given problem for problem solving Problem posing as reformulating a given problem for investigation Problem posing as constructing tasks for others

Each problem-posing activity was first illustrated using a mathematical miniature. Subsequently, findings on the respective problem-posing activities from the empirical research were summarized. In the end, it will be outlined that the notions of problem posing presented in this paper, as well as the summary of the empirical research state, should help researchers as well as practitioners in choosing the respective problem-posing activity in their studies of classrooms.

References Baumanns, L., & Rott, B. (2022). Developing a framework for characterizing problem-posing activities: A review. Research in Mathematics Education, 24(1), 28–50. https://doi.org/10. 1080/14794802.2021.1897036. Bonotto, C. (2013). Artifacts as sources for problem-posing activities. Educational Studies in Mathematics, 83(1), 37–55. https://doi.org/10.1007/s10649-012-9441-7. Bonotto, C., & Santo, L. D. (2015). On the relationship between problem posing, problem solving, and creativity in the primary school. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 103–123). Springer. Brown, S. I., & Walter, M. I. (1983). The art of problem posing. Franklin Institute Press. Brown, S. I., & Walter, M. I. (2005). The art of problem posing (3rd ed.). Lawrence Erlbaum Associates. Butts, T. (1980). Posing problems properly. In S. Krulik & R. E. Reys (Eds.), Problem solving in school mathematics (pp. 23–33). NCTM.

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Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing. Journal of Mathematical Behavior, 21(4), 401–421. https://doi.org/10.1016/S0732-3123(02)00142-6. Cai, J., & Hwang, S. (2020). Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research, 102, 1–8. https://doi.org/10.1016/j.ijer.2019.01.001. Cantor, G. (1867). De aequationibus secundi gradus indeterminatis. Schultz. Chen, L., Van Dooren, W., Chen, Q., & Verschaffel, L. (2011). An investigation on Chinese teachers’ realistic problem posing and problem solving ability and beliefs. International Journal of Science and Mathematics Education, 9(4), 919–948. https://doi.org/10.1007/ s10763-010-9259-7. Cifarelli, V. V., & Cai, J. (2005). The evolution of mathematical explorations in open-ended problem-solving situations. The Journal of Mathematical Behavior, 24(3–4), 302–324. https://doi.org/10.1016/j.jmathb.2005.09.007. Cifarelli, V. V., & Sevim, V. (2015). Problem posing as reformulation and sense-making within probem solving. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 177–194). Springer. Cruz, M. (2006). A mathematical problem-formulating strategy. International Journal for Mathematics Teaching and Learning, 7, 79–90. Duncker, K. (1945). On problem-solving. Psychological Monographs, 58(5), 1–113. Einstein, A., & Infeld, L. (1938). The evolution of physics: The growth of ideas from early concepts to relativity and quant. Cambridge University Press. Ellerton, N. F. (1986). Children’s made-up mathematics problems – A new perspective on talented mathematicians. Educational Studies in Mathematics, 17, 261–271. Hilbert, D. (1900). Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongreß zu Paris 1900. Nachrichten von der Königl. Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, 3, 253–297. Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–147). Lawrence Erlbaum Associates. Klinshtern, M., Koichu, B., & Berman, A. (2015). What do high school teachers mean by saying “I pose my own problems”? In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to effective practice (pp. 449–467). Springer. Koichu, B. (2020). Problem posing in the context of teaching for advanced problem solving. International Journal of Educational Research, 102, 101428. https://doi.org/10.1016/j.ijer. 2019.05.001. Koichu, B., & Kontorovich, I. (2013). Dissecting success stories on mathematical problem posing: A case of the Billiard Task. Educational Studies in Mathematics, 83(1), 71–86. https://doi.org/10.1007/s10649-012-9431-9. Kontorovich, I. (2020). Problem-posing triggers or where do mathematics competition problems come from? Educational Studies in Mathematics. https://doi.org/10.1007/s10649020-09964-1. Kontorovich, I., & Koichu, B. (2016). A case study of an expert problem poser for mathematics competitions. International Journal of Science and Mathematics Education, 14(1), 81–99. https://doi.org/10.1007/s10763-013-9467-z.

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Kontorovich, I., Koichu, B., Leikin, R., & Berman, A. (2012). An exploratory framework for handling the complexity of mathematical problem posing in small groups. The Journal of Mathematical Behavior, 31(1), 149–161. https://doi.org/10.1016/j.jmathb.2011.11.002. Lang, S. (1989). Faszination Mathematik – Ein Wissenschaftler stellt sich der Öffentlichkeit. Vieweg. Lee, S.-Y. (2021). Research status of mathematical problem posing in mathematics education journals. International Journal of Science and Mathematics Education, 19, 1677–1693. https://doi.org/10.1007/s10763-020-10128-z. Martinez-Luaces, V., Fernandez-Plaza, J., Rico, L., & Ruiz-Hildalgo, J. F. (2019). Inverse reformulations of a modelling problem proposed by prospective teachers in Spain. International Journal of Mathematical Education in Science and Technology, online. https://doi. org/10.1080/0020739X.2019.1683773. Nicol, C. C., & Crespo, S. (2006). Learning to teach with mathematics textbooks: How preservice teachers interpret and use curriculum materials. Educational Studies in Mathematics, 62(3), 331–355. https://doi.org/10.1007/s10649-006-5423-y. Padberg, F. (2008). Elementare Zahlentheorie (3rd ed.). Springer. Poincaré, H. (1973). Wissenschaft und Methode (unveränderter reprografischer Nachdruck der Ausgabe von 1914). Wissenschaftliche Buchgesellschaft. Pólya, G. (1957). How to solve it. A new aspect of mathematical method (2nd ed.). University Press. da Ponte, J. P., & Henriques, A. (2013). Problem posing based on investigation activities by university students. Educational Studies in Mathematics, 83(1), 145–156. https://doi.org/ 10.1007/s10649-012-9443-5. Poulos, A. (2017). A research on the creation of problems for mathematical competitions. The Teaching of Mathematics, 20(1), 26–26. Ramanujam, R. (2013). Paul Erd˝os. The artist of problem-posing. At Right Angles, 2(2), 5–10. Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press. https://www. sciencedirect.com/book/9780126288704/mathematical-problem-solving. Schoenfeld, A. H. (1989). Teaching mathematical thinking and problem solving. In L. B. Resnick & L. E. Klopfer (Eds.), Toward a thinking curriculum: Current cognitive Research (pp. 83–103). Association for Supervisors; Curriculum Developers. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition and sense – Making in mathematics. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). MacMillan. Schupp, H. (2002). Thema mit Variationen. Aufgabenvariationen im Mathematikunterricht. Franzbecker. Sharygin, I. F. (2001). The art of posing novel problems. Quantum, 8(2), 12–21. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28. https://flm-journal.org/Articles/2A5D152778141F58C1966ED8673C15. pdf. Silver, E. A. (2013). Problem-posing research in mathematics education: looking back, Looking around, and looking ahead. Educational Studies in Mathematics, 83(1), 157–162. https:// doi.org/10.1007/s10649-013-9477-3. Silver, E. A., Mamona-Downs, J., Leung, S., & Kenney, P. (1996). Posing mathematical problems: An exploratory study. Journal for Research in Mathematics Education, 27(3), 293–309. https://doi.org/10.2307/749366.

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Stoyanova, E. (1997). Extending and exploring students’ problem solving via problem posing (Doctoral dissertation). Edith Cowan University. http://ro.ecu.edu.au/cgi/viewcontent.cgi? article=1886&context=theses. Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing in school mathematics. In P. C. Clarkson (Ed.), Technology in mathematics education (pp. 518–525). Mathematics Education Research Group of Australasia. Tichá, M., & Hošpesová, A. (2013). Developing teachers’ subject didactic competence through problem posing. Educational Studies in Mathematics, 83(1), 133–143. https:// doi.org/10.1007/s10649-012-9455-1. Voica, C., & Singer, F. M. (2013). Problem modification as a tool for detecting cognitive flexibility in school children. ZDM – Mathematics Education, 45(2), 267–279. https://doi. org/10.1007/s11858-013-0492-8. Walter, M. I., & Brown, S. I. (1977). Problem posing and problem solving: An illustration of their interdependence. Mathematics Teacher, 70(1), 4–13. Xie, J., & Masingila, J. O. (2017). Examining interactions between problem posing and problem solving with prospective primary teachers: A case of using fractions. Educational Studies in Mathematics, 96(1), 101–118. https://doi.org/10.1007/s10649-017-9760-9. Ziegenbalg, J. (2014). Elementare Zahlentheorie. Beispiele, Geschichte, Algorithmen (2nd ed.). Springer.

Many Chefs in the Kitchen—a Collaborative Model for Problem-Posing Jason Cooper, Boris Koichu, Mirela Widder, Sarel Aiber, Yonah Amir, Aamer Badarneh, Menucha Farber, Michael Gorodin, Orly Gottlib, Esther Gruenhut, and Fatmeh Hihi 1

Introduction

Have you ever designed a challenging mathematical problem that you absolutely love, just to find that teachers hate it, that students don’t understand it, or that mathematicians find fault with it? It has long been recognized that diversity in development teams (e.g., Goldenberg, 1999), possibly organized as researchpractice design partnerships (Penuel et al., 2013), can help avoid such outcomes. Yet it remains unclear exactly how diversity can contribute productively to the design process of mathematical problems. The authors of this chapter comprise a diverse team of problem developers, including mathematics education researchers, teacher educators, and practicing teachers. We have joined forces in an R&D project (described henceforth) whose primary goal is to encourage middle school teachers, teaching high-track classes, to conduct an increasing portion of their lessons around challenging mathematical problems. In this chapter, we follow the development and revision of three problems in a process aimed to make these problems attractive and valuable in the eyes of all involved.

J. Cooper (B) · B. Koichu · M. Widder · S. Aiber · Y. Amir · A. Badarneh · M. Farber · M. Gorodin · O. Gottlib · E. Gruenhut · F. Hihi Science Teaching, Weizmann Institute of Science, Rehovot, Israel e-mail: [email protected] B. Koichu e-mail: [email protected] © The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 D. Sarikaya et al. (eds.), Problem Posing and Solving for Mathematically Gifted and Interested Students, https://doi.org/10.1007/978-3-658-41061-2_2

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We consider this activity—of developing and refining challenging mathematical problems—a case of problem posing (PP). While much of the research in this field is concerned with the activity of students posing problems, there is a growing body of research on teachers as problem posers (e.g. Baumanns & Rott, 2022). At the most basic level, this chapter aims to attend to the question that Kilpatrick (1987) put forth 35 years ago—where do good problems come from? Many of the PP mechanisms that Kilpatrick discussed (1987) are evident in the development team’s work, such as exploring mathematical and non-mathematical situations for inspiration, intentionally ill-structuring problems in an attempt to make them more realistic, reformulation of sub-problems and related problems (including generalization), seeking alternative formulations that lead to different mathematical models, and modifying conditions to obtain a more accessible problem to serve as a stepping stone when problems are found to be too challenging. With Kilpatrick’s question in mind, we set out to describe some of the principles that have come to guide our work, illustrate these principles through examples, and discuss how the team’s diversity contributes to the process and to the outcome. In this we are attending to many of the facets in Kontorovich et al.’s framework for handling the complexity of problem posing in small groups (2012), which include 1. Task organization (i. e. didactical decisions regarding the PP activity); 2. Solvers’ knowledge base (i.e. mathematical facts, definitions, procedures, competencies and prior experience); 3. Problem-posing heuristics and schemes (e.g. as put forth by Kilpatrick, 1987); 4. Group dynamics and interactions (normalization—convergence of PP norms, conformity—powerful team members attempting to make their perspective prevail, and innovation –new perspectives that emerge in attempts to resolve conflicts); 5. Individual considerations of aptness (comprehensions of explicit and implicit requirements of a problemposing task and relative importance of these requirements). In fact, all these facets are pertinent when considering a development team, with the exception of task organization (the task of problem-posing is not didactically organized for a development team).

2

The Project—Raising the Bar in Mathematics Classrooms

Raising the Bar in Mathematics Classrooms (RBMC) is an R&D program that aims to introduce challenging problem solving in high-track middle-school mathematics classes (grades 8–9, where the age of students is 13–15). To this end, we are developing a bank of challenging problems, providing teacher guides

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along with these problems, and leading professional development for teachers, undertaken as Professional Learning Communities (PLC, see Brodie, 2020). In the RBMC learning communities, teachers themselves solve problems, consider problem solving pedagogies that fit the context of their work, revise problems as they deem appropriate, enact them in their classrooms, and share their experience and insight with the community. The development and revision of problems is an ongoing process, advised by various sources, including: A. Teachers on the development team who try problems out in their own classrooms; B. Teacher educators leading the PLCs, who hear other teachers’ feedback, both before and after classroom experiences; C. Mathematicians, mathematics education researchers, teachers and ministry representatives on the program’s steering committee. Following a well-established mathematics education tradition (e.g., Schroeder & Lester, 1989), we view problem solving both as a goal in its own right (middle-school students should be able to apply the mathematics they have learned to authentic problem situations, OECD, 2019) and as a means toward a deeper understanding of curricular topics (e.g., Goldin, 2014). These two goals which we consider to be basically compatible, together with institutional expectations and constraints, shaped our work. It is beyond the scope of this chapter to detail in full the expectations of all the stakeholders in the project (but see Koichu et al., 2022), however we do note that the philanthropic organization that is funding the project (see Acknowledgement) seeks to promote high achieving students’ mathematical competencies around solving high-level realistic problems—levels 5–6 according to the PISA framework, OECD, 2019— while the Ministry of Education values a more general goal of engaging and promoting students at all levels in mathematical reasoning. Eventually, our work was guided by the following principles. • Cognitive/epistemic principles – Problems should encourage students to engage in mathematical reasoning, e.g., as conceived by the OECD in the PISA framework (OECD, 2019). – Problems should attend to important mathematical ideas that arise organically from a challenging situation (e.g., Charles & Carmel, 2005). • Affective principles – Problems should be realistic for students, in the sense that they can imagine the context and relate to it (Van den Heuvel-Panhuizen & Drijvers, 2020). – Problems should be engaging for students, attending to issues of a genuine interest for them (e.g., Goldin, 2009).

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– Problems should be attractive (e.g., innovative) in the eye of the designer (e.g., Kontorovich & Koichu, 2012). • Pedagogical/institutional principles – The mathematical content underlying problems should be connected to the mathematics curriculum (e.g., Sinclair & Jackiw, 2005). – Problems should engage students at different levels of mathematical proficiency and sophistication in classrooms that are inherently heterogeneous (e.g., Koichu, 2012; Mevarech & Kramarski, 1997). This list of principles is based on an amalgam of research (where referenced) and personal orientation. None of these principles are controversial, yet our work has revealed that implementing them is far from straightforward, and there may in fact be inherent tensions between them. We bring in this chapter some examples of our practical experiences in pursuing these principles. The team The members of the project team, all of whom are co-authors of this chapter, include four mathematics education researchers (BK, JC, MW, and MF), three teacher educators with experience teaching at education colleges and/or professional development programs (YA, OG, MW), and three practicing mathematics teachers (SA, MG, EG, MF). All of the researchers and all of the teacher educators have had some experience as mathematics teachers in schools, and the researchers all have some experience as teacher educators. The evidence The data for this chapter was elicited in an online meeting of the RBMC team, in which we reflected on the evolution of problems—how they were first conceived, what changes they underwent, and what these changes aimed to achieve. What follows is based on a summary and analysis of this discussion, relying on background documents that include interim versions of the problems and recordings of discussions and debates. The analysis Our analysis drew on Kontorovich et al.’s framework of problem posing in small groups (2012). For each problem, we first analyzed the initial version, attending to the problem poser’s knowledge base and considerations of aptness. We then analyzed dissatisfaction of some team members with the initial version, focusing on individuals’ divergent considerations of aptness. Finally, we analyzed the group

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dynamics of the revision process, seeking cases of normalization, conformity and innovation. Presenting the problems The exposition of all the problems follows a similar structure: • The initial version of the problem and what triggered its inception; • The dissatisfaction of some team members and subsequent revision; • Retrospective reflection

3

Problem 1—Jaime’s grade

3.1

The Initial Version of the Problem and What Triggered Its Inception

The development team’s search for PISA-style problems (see OECD, 2012 for examples of PISA test items) set in contexts that are familiar to students brought to BK’s mind a problem he had developed many years earlier, about a lazy student who wishes to invest minimal effort to achieve a passing grade (Fig. 1). The weights of 5 tests are given (12%, 24%, 12%, 12%, 40%), and the first two grades are known (80 and 60, respectively). The problem invites students to propose different ways to complete the table in Fig. 1. In its original form, the point of the problem was to demonstrate the heuristic strategy of working backwards—assume a particular final grade (e.g., a minimum passing grade of 56) and work backwards to propose ways to achieve it.

3.2

Revisions in Design and in Implementation

YA, an experienced teacher-educator, considered the problem attractive, recognizing that the situation could be investigated using the heuristic strategy of attending to extreme cases (the highest and lowest achievable final grades given the first two test scores). Furthermore, she saw in it a solid mathematical connection to linear programming—a topic from the lowest-track matriculation test for which she had developed curricular material years earlier. The student is in effect investigating linear combinations of 5 scores under a variety of constraints. This intra-mathematical connection created what YA described as a wow moment

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Fig. 1 Initial version of Jaime’s Grade problem

for her, revealing that the problem touches on some profound and essential topics in mathematics. Yet it was not immediately clear how to connect the problem to linear programming, which is not a part of the middle school curriculum, and thus cannot be referred to explicitly. It was also not obvious how to introduce such a connection in a manner that will arise organically in an authentic situation. YA also objected to the framing of the situation with respect to an unnamed lazy student. In YA’s hands this problem became Jaime’s grade for life.1 The beginning of the problem remained basically unchanged, the main revision being a new section that was added (see Fig. 2): In this section, students are invited to make sense of an unfamiliar representation, whereby points on a Cartesian line represent all combinations of two scores that will yield a particular final grade (80). Some researchers, including the late Judah Schwartz (personal communication with JC, January, 2018), see in making sense of unfamiliar representations the very essence of mathematical reasoning and sense-making. This is consistent with the PISA framework (OECD, 2019), where the flexible translation among different representations of information is one of the hallmarks of the highest level (6) of mathematical proficiency. 1

In Hebrew the word for life sounds like the name Jaime.

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Jaime decided to try out for the high-track class, for which he needs a final grade of 80. He worked hard and aced the third test with a score of 100 [...] The following “blind” graph (i.e., without indication of explicit numeric values) describes various outcomes for the final two tests, where the x-axis represents Jaime’s grade on the 4th test and the y-axis represents his grade on the 5th. Try to explain what the various parts of the sketch represent. In particular, what does the line whose equation is 36 + 0.12 + 0.4 = 80 represent, and where does the number 36 come from? What does the red triangle represent?

Fig. 2 Added section in Jaime’s Grade problem

MW, also an experienced teacher-educator, wished to bring this problem to a community of teachers she was leading, recognizing in it the potential to engage students with widely different mathematical abilities: some could apply trial and error to propose particular combinations of scores that achieve a desired final grade, while others could approach the problem systematically through symbolic manipulations of inequalities. However, the co-leader of her community, who is not part of the RBMC team, doubted the feasibility of the final section of the problem because, in her view, it deals with functions of two variables, a topic that is not part of the school curriculum. The discrepancy in views was not difficult to remediate: a principle that consistently guides our work is that all problems are presented in an editable format, and teachers are encouraged to adapt problems to their goals and educational contexts. Thus, it was possible to revise the problem to the satisfaction of both co-leaders. The section that appears to deal with a function of two variables was deleted from the problem text, and instead was discussed orally with teachers, signaling that it was appropriate for challenging the teachers, but not necessarily something that should be offered to students. OG and SA brought this problem to a community of teachers that they co-led. In this community the role of the table representation (see Fig. 1) changed. While it was originally introduced to support a trial-and-error strategy, appropriate for students who cannot or choose not to broach this problem symbolically, in this community the table was seen as one of four complementary representations of

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the problem.2 Thus, tabular data (see Fig. 1) supported making sense of the unfamiliar graphic representation (Fig. 2) by asking participating teachers to locate individual filled-in rows of the table as points on the graph. EG, a middle school mathematics teacher, enacted this problem in her own grade 9 class. Her students were highly engaged, yet for unanticipated reasons. Many of them disregarded the question as posed, and instead used the situation as a springboard to investigate the notion of weighted average, an activity that EG allowed and even encouraged. This highlights that even in a problem-solving context, students may choose to engage in problem-posing if this activity is valued and encouraged by the teacher.

3.3

Retrospective Reflection on the Problem and Its Development

The problem was first brought into the RBMC bank through BK’s personal history—it is a problem he had himself developed in the past and had good experiences with. Its main goal was to elicit the problem solving heuristic of working backwards. It was further developed when other team members recognized in it some potential for achieving a new goal that they have had some history with and found attractive—an inter-mathematical connection with linear programming, pedagogical support for classroom heterogeneity, or an opportunity for reasoning with representations. Thus, different team members, with different agendas, contributed to the problem in ways that attended to complementary considerations of aptness, and added new value without detracting from what others valued. The problem is striking for the difficulty in pinpointing the knowledge base that it relies on. It is related to problem-solving heuristics, to weighted average, to linear programming and to functions of two variables, yet does not require prior knowledge in any of these fields. While this was seen by some of the RBMC team as a strong point of the problem, it did create a tension for MW with a community co-leader who wished the mathematical content to be clearly specified and tightly integrated with the curriculum that teachers are required to teach in Israeli middle school. The lesson we learn from this is that given a problem that the developers value from many complementary perspectives, their goals and appreciation of the problem need not (and in fact cannot) be imposed 2

Yerushalmy (1999) has published curricular resources that target 4 representations of functions—verbal, numeric/tabular, graphic, and symbolic, and has investigated didactic affordances of linking these representations.

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in its implementation (see also Koichu et al., 2022). While these goals are clearly articulated in the teacher guide developed by the team as a resource to support implementation, those making use of the problem may re-interpret the problem for their own goals, stressing what they appreciate in it and omitting what they do not. The mathematical richness of this problem leaves much space for such re-interpretation. In its revised form, the problem is set in a situation that is familiar to students, or at least “realistic”—something they can imagine taking place in their own classroom (Freudenthal, 1991). Students with different levels of mathematical proficiency can approach the problem differently, interpreting the question of “what might Jaime’s scores have been” as an existence problem (list one or more examples through trial and error), or as exhaustive (investigate all possibilities for achieving a particular final grade, graphically and/or symbolically). Thus, all students in heterogeneous classrooms can experience success.

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Problem 2—Positive or False Positive?

4.1

The Initial Version of the Problem and What Triggered Its Inception

JC formulated the initial version of this problem as a response to a post on his social media—someone was tested positive after exposure to a confirmed covid19 carrier, yet the reliability of the result was questioned due to “reports that most of the positive tests are false-positive”. This post, which felt misleading and rather irresponsible, led him to delve into the mathematics of sensitivity and specificity of medical tests and the respective probabilities of false-negative and false-positive. The main mathematical point of the problem in its initial version was to attend to the common error of base-rate-neglect (e.g. Kahaneman, 2011). When the base-rate (i.e. the general prevalence) of a condition is very low, the overwhelming majority of positive tests are errors. For higher base rates, a lower percentage of positive tests are false-positive. Failing to realize this dependence on base rate is a common error called base-rate neglect. The problem aimed to show that there is no contradiction between low probability of a false result, given that the subject is healthy, and a high probability of a false result, given that the result is positive. Both refer to cases of false-positive. The first, which is called specificity, is a parameter of the medical test, while the second depends on the base rate of what is being tested. In the initial version of the problem (Fig. 3) a table was presented and students were instructed to calculate the probability

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of a positive test given that the subject is healthy, and the probability that Tami, who tested positive in a random screening test, is in fact healthy (section D). Both questions refer to cases of false-positive, however they are fundamentally different. The first takes the “whole” to be the healthy population, and thus relies 81 on data from the bottom row. The probability is 9795+81 = 0.82%. In the second, the “whole” is those who tested positive, and thus the relevant data is in the 81 = 39.7%. . In other words, the column on the right. The probability is 123+81 test is very reliable (high specificity, less than 1% probability of false-positive), yet nearly half the positive cases are testing errors! Students were asked to decide and explain which of the two represents the reliability of the test (section F). Before the problem was released, some team members felt that the notion of base-rate was not receiving sufficient attention. Hence a second table was introduced (Table 1), where sensitivity and specificity are similar to those in the first table (see Fig. 3), but the base-rate is much higher (9% of the population is sick, vs. 1% in the first table). The probability that a random positive test is in fact false-positive is now much lower (8% vs. 40% in the first table). Comparing

Fig. 3 Initial version of false-positive problem

Many Chefs in the Kitchen—a Collaborative … Table 1 Hypothetical test results with 1% base rate

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Test results of 10,000 randomly selected subjects Tested negative Tested positive 9

911

Found to have covid-19

8997

83

Found to be healthy

probabilities across these tables was intended to draw attention to what is different (their base-rate). This version of the problem (including both tables) addressed a well-known misconception in probability (base-rate-neglect) in a context that at the time (and for the foreseeable future) was realistic and highly relevant for students and for society at large. Students were invited to engage in high-proficiency mathematical reasoning related to a curricular topic (probability).

4.2

Revisions in Design and in Implementation

Revisions of the problem attended to a variety of concerns (see Appendix for revised version). Teachers has institutional concerns regarding aptness, feeling that the problem was too time-consuming. To address this, a shorter version was provided alongside the full version, where only the first table was given. Some team members had pedagogical concerns, feeling that the tables were difficult to make sense of, and that adding “total” data (the totals of each of the rows and of each of the columns) would be helpful. Others felt that calculating totals should be left for students. As a compromise, we added rows and columns titled total, leaving the data to be filled out by students (see Appendix). ON3 felt that students should be given the opportunity to construct frequency tables based on probability data (e.g., construct a table reflecting 99% sensitivity, 99% specificity, and 10% base rate). Such a section was added, but was later removed due to mathematical concerns (see henceforth). EG tried the revised problem out in her grade 9 class. She was gratified to find that the students enjoyed working on it and appreciated the novelty of a genuinely relevant problem. She further enhanced this sense of relevance by referring her students to an online newspaper article that discussed why, when covid-19 base 3

Previously a member of the development team.

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rate was low, authorities opted against screen testing (routine testing of seemingly healthy individuals to locate pre-symptomatic and non-symptomatic cases), because too many positive results would be false-positive. She then invited students to discuss connections between the article and the problem they had just solved. OF—a mathematician on the project’s advisory board—had the following concerns. An epistemic concern was that the problem might entrench two fundamental misconceptions. One is confounding experimental and theoretical probability; the probability of false-positive is inherently experimental, thus it is inappropriate to construct a table based on data that purports to reflect theoretical probability. The other is arguably even more fundamental; it is impossible to answer the question “what is the probability that Tami is sick” based on frequency data, because there is always some additional information about Tami that, when taken into consideration, influences this probability. How much time does she spend in public places? Does she wear a mask? Was she exposed to confirmed and/or symptomatic cases? Does she herself have symptoms? This mathematical concern resonated with an ethical concern: EG had an ethical concern—that students might conclude that positive tests can be ignored because they are usually errors, leading to socially irresponsible behavior. To attend to the first epistemic concern, we refrained from adding a section that asks students to construct a data table based on theoretical probabilities. The second epistemic concern was attended to along with the ethical concern in introducing a new section—a situation where three people tested positive, yet they were tested for different reasons. One was selected at random, one had been exposed to a confirmed case, and the third was showing covid-19 symptoms. In evaluating the likelihood that each of the results is false-positive, students are invited to consider the difference between random testing and “conditional” testing—given that the individual has been exposed to the disease and/or is showing symptoms. This revision created a tension for MG (a mathematics teacher), who felt that the pedagogical consideration outweighs the epistemic and societal considerations. In his words, “the essence of the mathematical insight (base-rate neglect) was lost, and the importance of base-rate was diluted for the sake of mathematical precision”. Furthermore, he felt that the misconception that the mathematician was concerned about, like many others in mathematics, is harmless enough in the context of secondary school, perhaps even unavoidable, and can be addressed if and when students take an undergraduate probability course.

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Retrospective Reflection on the Problem and Its Development

Once again, the development of the problem can be traced to team members’ personal, and even emotional engagement and considerations of aptness. Its inception was JC’s reaction to a post on a social media network that the author of the problem was troubled by. It was further developed after EG experienced her own students’ positive engagement and also found a strong thematic connection with an online article. The problem was first conceived with a very focused mathematical goal of highlighting a common misconception in probability referred to as base-rate neglect. Epistemic and societal concerns converged to suggest that the pure probabilistic consideration can be misleading (indicating that it’s “safe” to disregard positive tests), and even mathematically flawed (experimental data cannot determine the probability that a particular positive result is false). Thus, we opted to “dilute” the mathematical message with additional considerations that, while still mathematical, have a more qualitative nature (which of three positive results is most likely to be false). As a result, the problem became highly realistic for students, set in a familiar and relevant situation, and suggests societal implications that follow from mathematical reasoning. In one case, mathematical and societal considerations converged to suggest a revision. In another case, mathematical and pedagogical concerns diverged, and the lead developer of this problem (JC) opted for mathematical correctness over pedagogical affordances.

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Problem 3—The Relative Location of Median, Altitude, and Angle Bisector in Triangles

5.1

Impetus for Initial Version

Geometry textbooks pose countless problems where students are required to prove a claim. While proving often includes some problem-solving characteristics, most of the claims that students attempt to prove do not have intrinsic mathematical value (Herbst, 2002). OG came across such a problem in a textbook—proving that in a right-angled triangle, the right angle’s bisector bisects the angle between the median and the altitude. This fact is quite easy to prove relying on sums and differences of angles. However, it brought to mind a more general truth: for any vertex in any triangle, the angle bisector lies between the altitude and the median (and in fact coincides with them in the case of an isosceles triangle, see Fig. 4).

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Fig. 4 Median, angle bisector and altitude in right-angle triangle

Fig. 5 Initial version of segments in a triangle problem

To OG, this felt like something worth knowing and hence worth proving. This gave birth to the initial version of a problem (see Fig. 5). When this problem was shared among co-developers, YA contributed an applet that allows students to investigate the relative locations of these three lines. It was left to teachers to decide whether or not to provide an applet to assist students.

5.2

Revisions in Design and in Implementation

Initial revisions of the problem attended to a variety of concerns. Working with teachers in professional development raised a cognitive concern, when it became evident that the problem was more challenging than we had reckoned. In particular, it was difficult for them to model the claim of “betweenness”

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(e.g., as AM < AE < AH , or as k/2, there must be a row . R '' / = R such that both, the square in row . R '' and column .C and the square in row . R '' and column .C ' , contain a token. As there is more than one token in . R '' , it follows that . R '' must contain at

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Fig. 10 A configuration .C forcing .9 steps for .k = 8, .ℓ = 5

8 7 6 5 4 3 2 1 A

B

C

D

E

F

G

H

least .ℓ + 1 tokens. Now in total there are at least .4ℓ tokens in . R, R '' , C, C ' . As there are exactly .k + 2ℓ − 1 tokens in . D, it follows that .k + 2ℓ − 1 ≥ 4ℓ, a contradiction to .ℓ > k/2. Consequently, any row other than . R contains at most one token and, hence, the number of tokens in . D is at most .2k − 1. On the other hand, the number of tokens in . D is .2ℓ + k − 1 which again contradicts .ℓ > k/2. If.ℓ > k/2, then.k +2ℓ tokens suffice to force.k +1 steps as.k +1 steps are needed to resolve the configuration with .k tokens along a diagonal and .ℓ additional tokens in both, the first row and the first column. Figure 10 displays this configuration in the case .k = 8, .ℓ = 5, where we have 18 tokens for which nine steps are needed.

3.3

Discussion

Our discussion is structured in the following way. We will first argue why the teaching unit presented above is generally worthy from the perspective of mathematics education. Afterwards, each activity is considered separately and some brief notes regarding its purpose within the unit are given. The presented checkerboard problems meet the demands of teaching mathematical problem solving particularly well for several reasons. First of all, these problems combine substantial mathematical content with very simple playing rules and a minimum of required mathematical knowledge on the part of the students, which is widely accepted as essential for good problem solving material (see, e.g., Schoenfeld, 1985, p. 11) or even for teaching mathematics in general, as Wagenschein emphasizes:

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We should introduce as many problems as possible into mathematical teaching that combine a maximum of mathematical content with a minimum of needed routine [...]. (Wagenschein, 1970, p. 139)

Moreover, each of the checkerboard problems meets the definition of a mathematical problem (rather than an exercise task), as the students do not have easy access to a procedure for solving it (cf., Schoenfeld, 1985, p. 11). Another important point is that these problems are ‘nonstandard’ as Schoenfeld calls it (cf., Schoenfeld, 1985, p. 2), i.e., they are not covered in high school courses. This is advantageous, particularly for enrichment courses, as thereby everyone starts from zero on this topic, even in groups that are mixed in age or heterogeneous with regard to other aspects. As mentioned above, when the problem arose in mathematical research it was originally formulated in terms of lattice points, some of which were considered faulty. The reformulation in terms of tokens on a checkerboard serves several purposes. First of all, this enables the students to enactively work on the problems and actually play the checkerboard game with each other. Besides the mere activation of the students this encourages them to try things out and, especially when working in groups, ideas can be tested quickly and easily. Moreover, by working with physical material rather than iconic representations the cognitive load24 is reduced. This, together with the above mentioned fact that the rules are very simple, makes the checkerboard unit suitable for acquainting not only the highly gifted with problem solving. For more on the connection of Cognitive Load Theory and the teaching of problem posing for students who have average abilities in mathematics see Ambrus and Barczi-Veres (2016). The fact that the checkerboard problems are not only suitable for the highly gifted should not be misunderstood. They are fruitful for both students with average abilities and students with high abilities in mathematics. The weaker ones can approach the problems by trial and error and the stronger ones can find more elegant solutions. For instance, to give an exhaustive answer to Question 7 one has to argue that all tokens of a resolvable configuration with at most 20 tokens can be removed in at most 13 steps. On the one hand, it is possible and worthwhile to precisely prove this claim (as shown above in the solution to Question 7) with students who are capable to digest such a proof. On the other hand, it would be alright to content ourselves if the students only gain an understanding of this on the basis of systematic trial and error. Among the more elegant arguments that can be 24

Cognitive load can be defined as the load imposed on the working memory by presenting information (Ambrus & Barczi-Veres, 2016, p. 140). For a comprehensive introduction to Cognitive Load Theory see Sweller et al. (2011).

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useful to approach some of the checkerboard problems are double counting, indirect proof, and inductive arguments. Moreover, heuristic strategies, such as looking for patterns can be helpful. Note that these techniques are not requisite to successfully work on the checkerboard problems. In fact, the unit could be used to introduce them, or students with less experience in problem solving could learn them from more experienced students. Another strength of the presented topic is that it can be easily adjusted in difficulty and demand. This can be realized by considering the problems in greater or lesser generality with regard to aspects such as rules (how many tokens can be removed in each step?), complexity (boards of which size are considered?), or abstraction (do we content ourselves with dimension .2 or do we extend our considerations to the third or even arbitrary dimension?). Besides varying difficulty and demand one can decide whether formal notations, such as .s(C) or .t(s) should be used or avoided. As shown in the presentation of the activities it is easy to describe the meaning of these expressions by words, which would clearly be necessary when working with younger kids for instance. In concluding this section we want to briefly comment on some educational aspects of each activity. The first activity is thought to engage the students in actively exploring the rules and getting acquainted with notions that will be important in the further course of the unit, such as the least number of steps needed to resolve a given configuration or non-resolvable configurations. The example given in this activity ensures that the rules are at least once demonstrated to the students, and at the same time a practical form of notation for a sequence of steps is proposed. Note that the shown solution is not optimal (in the sense of using as few as steps as possible) for the sake of avoiding the students being mere recipients and to motivate them with the challenge of beating the given solution by resolving it with fewer steps. Such competitive situations are implemented again and again throughout the unit.25 In Activity 2 the notion of the smallest number .s(C) of steps needed to remove all tokens from a given configuration .C is substantiated by asking the students to find specific configurations .C for some small given values .s(C). It is helpful that they already determined .s(C) = 11 for the configuration .C in Activity 1 without knowing it. Moreover, the students find out what the largest possible value for .s(C) 25

To illustrate the momentum of motivation through competition we give an anecdote from a math camp in Japan in January 2020. A group of five girls were trying again and again to find a configuration with .21 tokens that forces .14 steps (in order to solve Question 7). Each time they came up with a proposal they called the second author (who was the instructor) to their table and asked him to resolve the configuration. While he was performing steps they all counted out loud together their number. He succeeded several times in a row to do it with only 13 steps, which was answered with screaming in a mixture of desperation and joy.

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is, which will be helpful in later activities. In Question 4 the answer to Question 3 is exploited to determine the largest possible number of tokens of a resolvable configuration and the students are asked to find a configuration with that number of tokens. In summary, this activity combines demanding general arguments on the one hand and specific examples on the other hand. As a result, students with different strengths and mindsets can contribute to solving these problems. The main aspect of Activity 3 is to force a given number of steps with as few tokens as possible. While this is pretty easy when the given number of steps is in .{1, 2, . . . , 8}, it gets more complicated for larger numbers of steps. Therefore, in order for the students to get a grasp of this kind of problem they are first asked to determine .t(1), . . . , t(8). The second part of Question 6, namely the question for .t(9) is a challenge that can call for teamwork and good bookkeeping or for clever arguments. Depending on the abilities of the group and the teaching intention one can focus on just finding a candidate for .t(9) and accepting it because the considered examples support the conjectured value, or one can focus on finding a precise justification. Question 7 asks for .t(14), and that is a good deal more challenging. But from our experiences most students will be highly engaged and motivated by now, so that they will feverishly search for suitable configurations. Activity 4 aims at instructors and learners who are keen to explore the topic in more depth and generality (and have enough time at their disposal). Making use of insights one has gained before, exploiting them by observing analogies, and generalizing arguments that have been developed earlier is core to this activity. Instructors who want to go even further with their group can consult the above mentioned references and will best know by themselves what questions should be pursued next. Depending on the age and background of the students, this unit can in addition be used to demonstrate a general aspect of mathematics: Namely, by explaining—or at least mentioning—at the end of the unit that the results of this playful occupation with checkerboard tokens is relevant for practical applications, namely, databases. This can serve as an example for the many instances in which mathematicians studied a topic of “pure” mathematics—such as elementary number theory—for its own sake, beauty, interest or simply out of playfulness and much—perhaps centuries—later, their results found applications they would not have dreamed of (such as the use of number theory in cryptography). In many of these instances, it is hardly conceivable that the underlying theory would have been found by someone merely interested in and oriented towards the respective application. Thus, even someone who believes that the ultimate justification of science lies in its (technical)

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applications has a reason to tolerate, respect and encourage intellectual playfulness. It would be a great success if students leave the unit with such an encouragement.26

4

Elementary Length Formulas for Triangles and Quadrilaterals

Merlin Carl, Uwe Leck, Hinrich Lorenzen, Michael Schmitz

4.1

Introduction

In this article we present a geometric teaching unit suitable for the use in enrichment activities for interested and talented high school students, but not only for the highly gifted. The unit, in particular some of the solutions, may look somewhat unusual from the perspective of school geometry. For instance, we deliberately avoid angles and trigonometry, such that some of the proofs may look more complicated than necessary. On the other hand, there is a lot to gain using our approach: • We do not need to distinguish any cases, as our arguments are independent of the mutual arrangement of the different geometric objects. • Our approach is consistent in that the same line of argumentation is used throughout, i.e., we build a little theory. • We prepare the ground for working with more abstract geometric models used in higher mathematics. The unit was delivered in spring 2021 at an online academy with 31 participants of ages from 16 to 18. More information is given in our paper (Carl et al., 2023c) in this volume. The presentation in Sects. 4.2 and 4.3 is meant as an outline of a teaching implementation that is structured along problems, hints, and solutions. The reader should feel free to alter the course of investigation according to their learning group. Problems can be added or omitted, the same applies to hints. The hints formulated here are only sketchy suggestions to give an impression what suitable help could look like. Although the presented unit is geometric, it also fosters skillful and purposeful 26

We note, however, that in this particular case, the actual order of discovery was the other way round: As mentioned in the introduction, the second author hit upon the checkerboard combinatorics when thinking about databases. Even in such cases, the playful occupation can lead to (i) a more thorough-going investigation and to questions that would have been less natural on the basis of the intended application alone, but can still be quite relevant for applications and (ii) a way of representing the problem in such a general and abstract way that one can now observe that other application problems are also instances thereof.

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term manipulation. Among other things, this aspect is discussed in greater detail in Sect. 4.4, where thoughts from the perspective of mathematics education are shared. More information on the motivation and the special focus of the unit can also be found there. The impatient reader is encouraged to read through Sect. 4.4 in advance. Note that all formulas that we derive in this article are applicable to arbitrary triangles or quadrilaterals, unless additional conditions are stated explicitly. That means, for instance, that quadrilaterals are allowed to be self-intersecting, or collinear, or that vertices may coincide. A corresponding system of axioms as well as length formulas involving circles can be found in Lorenzen (2003).

4.2

Length Formulas for Triangles

We denote points by capital letters. The straight line through two given points is denoted by . X Y and the line segment between . X and .Y by . X Y . To address a triangle with vertices. X , Y , Z we write. X Y Z . We assume that Pythagoras’ Theorem is well-known, and we start with recalling some closely related results concerning right triangles, namely the Altitude Theorem, the Cathetus Theorem, and another theorem that we call the Area Theorem. Throughout, we will label triangles in the usual way, i.e., if. ABC is a triangle we denote the lengths of the sides opposite to the points . A, B, C by .a, b, c, respectively. Moreover, if . ABC has a right angle at .C and . L is the foot of the altitude with respect to .C, we denote the length of the altitude by .h and the lengths of the line segments . AL and . L B by . p and .q, respectively. If . ABC is not a right triangle, we still denote the length of the altitude with respect to .C by .h, and the segment lengths .|AL| and .|L B| by . p and .q, respectively. Note that this choice of notation is a bit unusual. We do not always have .c = p + q, but we have .c = p + q or . p = c + q or .q = c + p, see Fig. 11. It will be explained later why this is sufficient (and convenient). .X , Y

Fig. 11 Standard notations for arbitrary triangles

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Using this notation, the following identities hold for all triangles with a right angle at .C. Altitude Theorem (AlT): .h 2 = pq Cathetus Theorem (CT): . pc = b2 and .qc = a 2 Area Theorem (ArT): .hc = ab Depending on whether these results and their proofs are known by the students, the first problem can be used or omitted. Problem 1 Prove the Altitude Theorem, the Cathetus Theorem, and the Area Theorem (in this order). Solution Applying Pythagoras’ Theorem to. ABC yields.a 2 +b2 = c2 , and applying it to . ALC and . L BC we obtain . p 2 + h 2 = b2 and .q 2 + h 2 = a 2 (see Fig. 12). Moreover, we have . p + q = c. Hence, .c

2

= p 2 + q 2 + 2 pq = b2 − h 2 + a 2 − h 2 + 2 pq = c2 − 2h 2 + 2 pq.

Rearranging gives .h 2 = pq, i.e., (AlT) is proven. Furthermore, we have . pc

= p( p + q) = p 2 + pq = p 2 + h 2 = b2 . (AlT)

The second part of (CT) can be shown analogously. Finally, 2 2 .h c = pqc2 = (qc)( pc) = a 2 b2 . (AlT)

(CT)

Taking the roots on both sides completes the proof of (ArT).

Fig. 12 Standard notations for right triangles

☐

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We are now heading for generalized versions of the above considered formulas, which hold for arbitrary triangles. To this end, we consider the following function of three real variables: . F(x,

y, z) := 4x y − (x + y − z)2 .

The statement in the following problem will be useful later. Problem 2 Show that . F is totally symmetric, i.e., . F(x, y, z) does not depend on the order of the arguments .x, y, z. Solution We just rewrite the expression given in the definition of . F: .4x y

− (x + y − z)2 = 4x y − (x 2 + y 2 + z 2 + 2x y − 2x z − 2yz) = 2x y + 2x z + 2yz − (x 2 + y 2 + z 2 ).

Obviously, the latter expression is symmetric in .x, y, z.

☐

In what follows, it will be crucial to consider expressions of the form . F(a 2 , b2 , c2 ), and it is easily seen that these can be written in the form . F(a

2

, b2 , c2 ) = (a + b + c)(a + b − c)(b + c − a)(c + a − b).

(1)

Problem 3 Explain why formula (1) holds, without actually expanding the product on the right-hand side. Solution We have to show that ) ( = 2 a 2 b2 + a 2 c2 + b2 c2 −(a 4 +b4 +c4 ). (2) When expanding the product on the left-hand side, we have to choose one summand from each of the four parentheses, multiply them, and add up these product over all possible choices of the four summands. That means, we have to choose exactly one value from each row of the following table.

.(a+b+c)(a+b−c)(b+c−a)(c+a−b)

a b c a b −c . a −b c −a b c

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If we choose the same column in each row, we get an expression of the form .−x 4 with .x ∈ {a, b, c}. This explains the term in the latter parentheses on the right-hand side in (2). To understand how the term in the other parentheses comes about, we observe that if we choose twice from each of two columns only, we get an expression of the form .4x

y − 2x 2 y 2 = 2x 2 y 2

2 2

with x, y ∈ {a, b, c} and x / = y.

It remains to explain that the other choices (i.e., those that use all three columns) lead to expressions that cancel out. This can be done similarly. Every expression of the form .x 2 yz or .−x 2 yz with distinct .x, y, z ∈ {a, b, c} arises exactly six times. ☐ From (1) it follows that . F(a 2 , b2 , c2 ) = 0 if and only if .a, b, c form a sum, i.e., if .a +b = c or.a +c = b or.b+c = a. This is particularly interesting when considering three points on a straight line and the lengths of the segments they form, such as . p, q, c in the right triangles considered above. The result in the following problem is an important next step for our further investigations, and it should be noted that it holds for arbitrary triangles. Moreover, note that we do not need .c = p + q, but only that . p, q, c form a sum, which is true in arbitrary triangles if . p, q, c are defined as above. As a consequence, our proofs work without distinguishing cases, and this is an advantage over usual approaches, where . p and .q are defined to be “signed lengths”, i.e., . p = −|AL| if . ABC has an obtuse angle at . A and .q = −|B L| if . ABC has an obtuse angle at . B. Problem 4 Let . ABC be an arbitrary triangle with side lengths .a, b, c. Let . L be the foot of the altitude with respect to .C and .h = |LC|. Prove the Altitude Formula (AF) 2 2 2 2 2 .4h c = F(a , b , c ). Hint: Let . p = |AL| and .q = |L B|. Consider . F( p 2 , c2 , q 2 ) and substitute . p 2 and 2 .q . Solution It might be helpful to consider the right-hand side of Fig. 11. By Pythagoras’ Theorem, we have . p 2 + h 2 = b2 and .q 2 + h 2 = a 2 . As mentioned above, since 2 2 2 2 2 2 2 2 2 . p, c, q form a sum, . F( p , c , q ) = 0. Hence, we obtain .4 p c = ( p + c − q ) . Thus, 2 2 2 2 2 2 2 2 2 .4(b − h )c = (b − h + c − (a − h )) . It follows that .4h 2 c2 = 4b2 c2 − (b2 + c2 − a 2 )2 = F(a 2 , b2 , c2 ).

☐

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This enables us to derive a generalized version of the Cathetus Theorem. Problem 5 Let . ABC be an arbitrary triangle with the same notations as in Problem 4. Prove the Generalized Cathetus Theorem (GCT) . pc

=

1 2 1 |b + c2 − a 2 | and qc = |a 2 + c2 − b2 |. 2 2

Hint: Start with . p 2 c2 = . . . and make use of the Altitude Formula. Solution We refer to Fig. 11. We have .p

2 2

c

= (b2 − h 2 )c2 = b2 c2 − h 2 c2

Pyth.

= b2 c2 −

(AF)

=

1 F(a 2 , b2 , c2 ) 4

1 1 (4b2 c2 − F(a 2 , b2 , c2 )) = (b2 + c2 − a 2 )2 . 4 4 ☐

Problem 6 The usual Cathetus Theorem states that in a right triangle we have = b2 and .qc = a 2 . Show that the Cathetus Theorem does indeed follow from the Generalized Cathetus Theorem (which justifies its name).

. pc

Solution In a right triangle, we have .c2 = a 2 + b2 , due to Pythagoras. Hence, the Generalized Cathetus Theorem yields . pc

1 2 1 1 |b + c2 − a 2 | = |b2 + (a 2 + b2 ) − a 2 | = |2b2 | = b2 2 2 2

=

and, similarly, .qc

=

1 2 1 1 |a + c2 − b2 | = |a 2 + (a 2 + b2 ) − b2 | = |2a 2 | = a 2 . 2 2 2 ☐

Problem 7 Let . ABC be an arbitrary triangle. Using the same notation as in Problem 4, let .ρ = bp . Prove that .a 2 = b2 + c2 − 2bcρ.27

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Fig. 13 Notations used in the Four-Point Formula

Solution By the Generalized Cathetus Theorem, we have . pc

=

1 2 |b + c2 − a 2 |. 2

(∗)

We assume that . ABC is acute, the other case being analogous. Then we have .a 2 < b2 + c2 , such that .|b2 + c2 − a 2 | = b2 + c2 − a 2 . Hence, (.∗) is equivalent to p 2 2 2 2 2 2 .2 pc = b +c −a and, thus, to.a = b +c −2 pc. Noting that. pc = b· ·c = bcρ, b we obtain the desired result. ☐ The result in the following problem is even more general. It applies to arbitrary triangles, and the altitude with respect to .C is replaced by a line segment from .C to an arbitrary point on the base side of the triangle. It covers the Altitude Theorem as a special case. Problem 8 Let . ABC be an arbitrary triangle, and let . L be some point on the line segment . AB. Furthermore, let .u = |AL|, .v = |B L|, and .ℓ = |LC| (see Fig. 13). Prove the Four-Point Formula (FPF): .ℓ

2

=

u 2 v 2 a + b − uv. c c

Hint: Apply the Altitude Formula twice, once on . ABC and once on . L BC.

Solution The Altitude Formula applied to . ABC and . L BC yields 27

Noting that .ρ = cos(α), this is a non-trigonometric version of the law of cosines.

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F(a 2 , v 2 , ℓ2 ) F(a 2 , c2 , b2 ) 2 = h = . 4c2 4v 2

Using the definition of . F, we see that this implies .

a 2 + c2 − b2 a 2 + v 2 − ℓ2 = . c v

This can be rearranged to .ℓ

2

= a2 + v2 −

v(a 2 + c2 − b2 ) . c

The expression on the right-hand side equals ( .

1−

v) 2 v 2 u v a + b + v 2 − vc = a 2 + b2 − uv, c c c c

where we used .c = u + v.

☐

Equipped with what we have derived so far, we can obtain a nice formula for the radius of the circumcircle of an arbitrary triangle. Problem 9 Let . ABC be an arbitrary triangle, and let .r be the radius of its circumcircle. Show that ab abc .r = . = √ 2h F(a 2 , b2 , c2 ) Hint: Make use of the Altitude Formula and the Four-Point Formula. Solution Let . M be the midpoint of . AB, and let . X be the foot of the altitude with respect to . AB. Furthermore, let .U be the center of the circumcircle, and let .Y be the foot of the perpendicular to .C X through .U . We use the notations .x = |M X |, . y = |U M|, and .s = |MC| (see Fig. 14). Then we have (1) .|X Y | = y, (2) .|U Y | = x. By Pythagoras’ Theorem,

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Fig. 14 Notations in the proof of the formula for the radius of the circumcircle

(3) .s 2 = x 2 + h 2 , 1 (4) .r 2 = y 2 + c2 , 4 (5) .r 2 = |U Y |2 + |Y C|2 . Moreover, (6) . F(h 2 , |X Y |2 , |Y C|2 ) = 0. We obtain .|X Y |

2

1 = y 2 = r 2 − c2 , (1) (4) 4

and .|Y C|

2

= r 2 − |U Y |2 = r 2 − x 2 = r 2 − s 2 + h 2 .

(5)

(2)

(3)

Plugging this into (6) yields ( ) ( ) ( ) ) 2 ( 2 1 2 1 2 2 2 2 2 2 .4h . r − c = h + r − c − r −s +h 4 4 2

It follows that .4h

( )2 ( ) )2 1( 2 1 1 2 1 2 1 2 2 a + b2 − c2 . = = r −h 2 c2 = s 2 − c2 a + b − c (FPF) 2 4 2 2 4

2 2

Applying the Altitude Formula, we obtain

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)2 1( 2 a + b2 − c2 4 )2 1 1( 2 a + b2 − c2 = a 2 b2 . = F(a 2 , b2 , c2 ) − 4 4

r = h 2 c2 −

2 2

☐ The next problem and Problem 12 demand less complicated derivations, but ask the students to find formulas by themselves. Problem 10 Find a formula for the length of a median of a triangle . ABC that depends only on .a, b, c. Solution Let . M be the midpoint of . AB and .ℓ = |MC|. Then, using the notation from above, we have .u = v = 2c and the Four-Point Formula yields .ℓ

2

=

1 2 1 2 1 2 a + b − c . 2 2 4

(3) ☐

The next problem28 is interesting in its own right and will be useful for the solution to Problem 12. Problem 11 Let . ABC be an arbitrary triangle, and let . X be some point on . AB with . X / = A, B. Show that .C X is an inner angle bisector with respect to .C if and |AX | |AC| 29 only if . |B X | = |BC| . Solution Let .Y be the point of intersection of . BC and the straight line through . X that is parallel to . AC (see Fig. 15). Then by intercept theorems, we have .

28

|AX | |CY | = |B X | |BY |

(4)

The statement in this problem is a theorem from elementary geometry related to the famous Apollonius Theorem. For an alternative proof to the one presented here see, e.g., Alsina and Nelsen (2015, p. 66). 29 If instead . X is on . AB but not on . AB, then .C X is an outer angle bisector with respect to .C |AX | |AC| if and only if . |B X | = |BC| , provided that .|AC| / = |BC|. This can be shown analogously.

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Fig. 15 . X Y is parallel to . AC

and .

|AC| |X Y | = . |BC| |BY |

(5)

Moreover, let .Y ' be the image of .Y under reflection in .C X . Then . X Y CY

'

is a kite.

(6)

Now, the following equivalences hold. C X is an inner angle bisector with respect to C. ⇔ Y ' lies on AC. ⇔ X Y is parallel to CY ' . ⇔ X Y CY ' is a trapezoid. .

(6)

⇔ X Y CY ' is a rhombus. ⇔ |X Y | = |CY | (4) |AX | |B X | (5) |AX | ⇔ |B X|

⇔

The proof is complete.

= =

|X Y | |BY | |AC| |BC|

☐

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The next problem asks for a formula of the length of an inner angle bisector of a triangle. An alternative solution to the one presented here can be found in Alsina and Nelsen (2015) on p. 68. Problem 12 Find a formula for the length of an inner angle bisector of a triangle that depends only on .a, b, c.

. ABC

Solution Let .w be the inner angle bisector with respect to .C and let . L be the point of intersection of .w and . AB. Furthermore, let .ℓ = |C L|, .u = |AL|, and .v = |L B|. By Problem 11, we have . uv = ab since .w is an angle bisector. This implies .

and . vc =

a a+b .

u u 1 = = c u+v 1+

v u

=

1 1+

a b

=

b , a+b

Now the Four-Point Formula implies .ℓ

2

= = = = =

u 2 v 2 a + b − uv c c b a a2 + b2 − uv a+b a+b ab − uv cb ca ab − · a+b a+b ( ( )2 ) c . ab 1 − a+b ☐

4.3

Length Formulas for Quadrilaterals

In this section, we exploit our insights about triangles to find length formulas for quadrilaterals. For a quadrilateral . ABC D, we denote the lengths of the sides in the usual manner, i.e., .a = |AB|, .b = |BC|, .c = |C D|, .d = |D A|. Further, we denote the length of the diagonals by .e = |AC| and . f = |B D|. The result in the first problem of this section is central for what follows.

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Problem 13 Let . ABC D be an arbitrary quadrilateral and let .m be the distance between the midpoints of its diagonals. Prove the Midpoint-Distance Theorem (MDT) ) 1( 2 2 a + b2 + c2 + d 2 − e2 − f 2 . .m = 4 In particular, .a 2 + b2 + c2 + d 2 − e2 − f 2 ≥ 0. Hint: Apply formula (3) for the length of a median from Problem 10 to three suitable triangles. Solution Figure 16 illustrates the situation. Using the notations as in the figure, we derive the formula as follows. The line segment . B M is the median of . AC B with respect to . B. By (3) we obtain .|B M|2 = 1 2 1 2 1 2 1 2 1 2 1 2 2 2 a + 2 b − 4 e . Analogously, we have .|D M| = 2 c + 2 d − 4 e . The line segment . M L is the median of . B D M with respect to . M. Thus, (3) yields .m

2

= |M L|2 =

1 1 1 |B M|2 + |D M|2 − f 2 . 2 2 4

Plugging in the above expressions for .|B M|2 and .|D M|2 gives the MidpointDistance Theorem. ☐

Fig. 16 Sketch for the proof of the Midpoint-Distance Theorem

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Equipped with this result, we can easily derive a series of characterizations for certain types of quadrilaterals. Problem 14 (Parallelogram Characterization) Let . ABC D be a quadrilateral. Show that . ABC D is a parallelogram if and only if .a 2 + b2 + c2 + d 2 = e2 + f 2 .

Solution We use the well-known fact that . ABC D is a parallelogram if and only if its diagonals meet at their common midpoint (for a proof see, e.g., Kiselev, 2006, p. 70). By this, .m = 0 if and only if . ABC D is a parallelogram, and the claim immediately follows by the Midpoint-Distance Theorem. ☐

Problem 15 (Rectangle Characterization) Let . ABC D be a quadrilateral. Show that . ABC D is a rectangle if and only if .a 2 + b2 + c2 + d 2 = 2e f .

Solution We use the well-known fact that . ABC D is a rectangle if and only if its diagonals have the same length and meet at their common midpoint (for a proof see, e.g., Kiselev, 2006, p. 72). Then the following equivalences hold. .

ABC D is a rectangle. ⇔ ABC D is a parallelogram and e = f . ⇔ a 2 + b2 + c2 + d 2 − e2 − f 2 = 0 and (e − f )2 = 0 ⇔ a 2 + b2 + c2 + d 2 − e2 − f 2 + (e − f )2 = 0 ⇔ a 2 + b2 + c2 + d 2 = 2e f

In the second to last step we used the fact that for non-negative reals .x, y we have = 0 and . y = 0 if and only if .x + y = 0 (note that .a 2 + b2 + c2 + d 2 − e2 − f 2 ≥ 0 according to the Midpoint-Distance Theorem). We will use this argument again in ☐ the following characterizations.

.x

Problem 16 (Rhombus Characterization) Let . ABC D be a quadrilateral. Show that . ABC D is a rhombus if and only if .a 2 + b2 + c2 + d 2 = ab + bc + cd + da. Solution The following equivalences hold.

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ABC D is a rhombus. ⇔a=b ∧ b=c ∧ c=d ∧ d=a ⇔ (a − b)2 = 0 ∧ (b − c)2 = 0 ∧ (c − d)2 = 0 ∧ (d − a)2 = 0 ⇔ (a − b)2 + (b − c)2 + (c − d)2 + (d − a)2 = 0 ⇔ a 2 + b2 + c2 + d 2 = ab + bc + cd + da ☐

Problem 17 (Square Characterization) Let. ABC D be a quadrilateral. Show that 2 +b2 +c2 +d 2 +e2 + f 2 = ab+bc+cd+da+2e f .

. ABC D is a square if and only if.a

Solution The following equivalences hold. .

ABC D is a square. ⇔ ABC D is a rhombus and e = f . ⇔ a 2 + b2 + c2 + d 2 − ab − bc − cd − da = 0 and (e − f )2 = 0 ⇔ a 2 + b2 + c2 + d 2 − ab − bc − cd − da + (e − f )2 = 0 ⇔ a 2 + b2 + c2 + d 2 + e2 + f 2 = ab + bc + cd + da + 2e f

Note that we need .a 2 + b2 + c2 + d 2 − ab − bc − cd − da ≥ 0 in the second to last step. This is true since .a 2 + b2 + c2 + d 2 − ab − bc − cd − da = 21 [(a − b)2 + ☐ (b − c)2 + (c − d)2 + (d − a)2 ]. In the following, we say that a quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides. For convenience, we always assume that in a trapezoid . ABC D the sides . AB and .C D are parallel. The reason for this restriction is that the characterizations below can be stated more easily when the parallel sides are fixed. Problem 18 Let . ABC D be a convex quadrilateral with the usual notations. Then we have .4m 2 ≥ (a − c)2 , and equality holds if and only if . ABC D is a trapezoid. Solution It is helpful to consider Fig. 17. The Triangle Midsegment Theorem applied to . AB D yields .|K N | = a/2, and applied to . AC D it yields .|K M| = c/2. The triangle inequality applied to. K N M gives.|K M|+m ≥ |K N | and.|K N |+m ≥ c−a |K M|. Thus, we obtain.m ≥ |K N |−|K M| = a−c 2 , and.m ≥ |K M|−|K N | = 2 , i.e., .4m 2 ≥ (a − c)2 .

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Fig. 17 A convex quadrilateral

We now turn to the characterization of trapezoids. By the Midsegment Theorem, applied to the same triangles as above, we obtain. K M || C D and. K N || AB. Hence, . ABC D is a trapezoid if and only if . K M || K N , and this is the case if and only if . K , M, N are collinear. Since . ABC D is convex, . M and . N lie on the same side of . AD. Thus, . K cannot lie between . M and . N , and therefore . K , M, N are collinear if and only if .|K M| + |M N | = |K N | or .|K N | + |M N | = |K M|. Since we know that .{|K M|, |K N |} = {a/2, c/2}, this is equivalent to .m = |a−c| 2 , and the proof is complete. ☐ Problem 19 Let . ABC D be a convex quadrilateral with the usual notations. Show that . ABC D is a trapezoid if and only if .b2 + d 2 + 2ac = e2 + f 2 . Solution The following equivalences hold, where we used the Midpoint-Distance Theorem in the second to last step. .

ABC D is a trapezoid. ⇔ 4m 2 = (a − c)2 ⇔ a 2 + b2 + c2 + d 2 − e2 − f 2 = (a − c)2 ⇔ b2 + d 2 + 2ac = e2 + f 2 ☐

Problem 20 Let . ABC D be a convex quadrilateral with the usual notations. Show that . ABC D is an isosceles trapezoid if and only if .b2 + d 2 + 2ac = 2e f . Solution The following equivalences hold.

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ABC D is an isosceles trapezoid. ⇔ ABC D is a trapezoid and e = f . ⇔ b2 + d 2 + 2ac − e2 − f 2 = 0 and (e − f )2 = 0 ⇔ b2 + d 2 + 2ac − e2 − f 2 + (e − f )2 = 0 ⇔ b2 + d 2 + 2ac = 2e f

In the second to last step we used that.b2 +d 2 +2ac −e2 − f 2 ≥ 0. To see that this is the case, note that by Problem 18 we have.4m 2 ≥ (a −c)2 , i.e.,.2ac ≥ a 2 +c2 −4m 2 . This yields .b

2

+ d 2 + 2ac − e2 − f 2 ≥ a 2 + b2 + c2 + d 2 − e2 − f 2 − 4m 2 = 0,

where we used that by (MDT) we have .4m 2 = a 2 + b2 + c2 + d 2 − e2 − f 2 . ☐ If the learning group is keen to proceed, further characterization formulas—for kites and for symmetric kites, for instance—can be found.

4.4

Discussion

In this section, we briefly discuss the unit presented above from the perspective of mathematics education. Note that we strongly recommend to let the students use GeoGebra30 or a similar dynamic geometry software for explorations throughout the unit. This is helpful in several ways, such as having a tool for quickly checking whether conjectured formulas hold true, gaining a better geometric understanding via visualization, and drawing motivation out of “seeing that the discovered formulas actually work”. Obviously, the main educational goal of the presented unit is to impart a little geometric theory. As mentioned above, we follow a somewhat unusual path to gain the merits listed in the introduction. Moreover, the results we derive provide a list of formulas that are useful in a range of problems about triangles and quadrilaterals. This makes the unit beneficial for gifted students who want to participate in mathematical competitions or just want to extend their geometrical toolbox. Increasing the students’ abilities in purposeful term manipulation on an elementary level is a side benefit of the presented unit. It is widely accepted that working with algebraic expressions is a big obstacle for learners in mathematics (cf. Vollrath, 30

www.geogebra.org

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1993, pp. 86–88). Consequently, much effort is put into teaching and practicing term manipulations in school. Considering the complexity of algebraic expressions and equations in school textbooks and central examinations, one gets the impression that students who succeeded in school should have strong algebraic abilities at their disposal. Nevertheless, experiences from beginner courses at university are in stark contrast to this assumption. It is observed that many students have major difficulties even with very basic term manipulations (cf., e.g., Malle, 1993). Malle (1993) argues that one reason, among others, for this contradiction is the fact that algebra instruction tends to proceed too quickly from elementary to more complicated algebraic expressions, so that the students are deprived of the opportunity to internalize basic principles. Consequently, he recommends to spend more effort on working seriously, i.e., in a creative and purposeful way, with less complicated elementary algebraic expressions. The above presented unit offers some opportunities to follow this recommendation, which we obviously strongly support. The term “purposeful” should not be confused with a call for real-life references. It is meant to express that in the teaching unit presented here, the difficulty often lies in understanding where a certain term manipulation will lead to, rather than in performing it. This teaching unit also nicely illustrates the divergent aspects of mathematical thinking. In most of the cases, it will be difficult to immediately see the way to a solution to the very end. Rather, one tries to construct formulas that establish some connection between the relevant data, and then sees where this leads to. It can thus be used to show how free exploration is often of value, even when working on a very concrete problem. The unit can also serve as a demonstration of the stepwise construction of a mathematical theory. Although courses in school will often build on each other, this process is usually too “slow” to be well-visible for students. Here, however, one can see well how each new result at the same time is a step upon which the following results rest and which makes it heuristically much easier to prove them. At the same time, the unit shows the power of an algebraization without going all the way to calculating with coordinates: far from being a mere notational convenience, the algebraic representation offers powerful possibilities for manipulations and deductions, and thus displays a power that goes way beyond a mere symbolic rewriting. It is exactly that interaction of geometry and algebra that enables progress, and the unit provides an opportunity to point this out. One possibility to present the material in this paper is as a series of increasing generalizations: We start with Pythagoras’ Theorem, which is probably well-known to the students, along with the theorems in its “vicinity”, such as the Cathetus Theorem. Pythagoras’ Theorem is about right triangles, so we ask whether something similar can be proved for arbitrary triangles. Indeed, the Altitude Formula is such

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a generalization, and the generalized Cathetus Theorem shows that the “vicinity” generalizes along. One can then ask whether a further generalization is possible by generalizing the height to an arbitrary line from .C to some point on . AB, which is answered by the Four-Point Formula. Thus, these results can be seen as growing out of Pythagoras’ Theorem by continued relaxation of the assumptions. The further results on triangles can then be taken as a demonstration how fruitful such generalizations are. This would be an example of “free exploration”, following the question “What else can we do with this?”. Another possible approach would be to start right away with the question whether there is a law connecting the relevant lengths in an arbitrary triangle with a line connecting .C to some point of . AB. Pythagoras’ Theorem could then be motivated as a first answer in a very special case, which is then further developed with the goal in sight, providing an example of “directed exploration”.

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