Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research PDF
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ICME-13 Monographs Eric W. Hart James Sandefur Editors Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research ICME-13 Monographs Series editor Gabriele Kaiser, Faculty of Education, Didactics of Mathematics, Universität Hamburg, Hamburg, Germany Each volume in the series presents state-of-the art research on a particular topic in mathematics education and reflects the international debate as broadly as possible, while also incorporating insights into lesser-known areas of the discussion. Each volumeisbasedonthediscussionsandpresentationsduringtheICME-13Congress and includes the best papers from one of the ICME-13 Topical Study Groups or Discussion Groups. More information about this series at http://www.springer.com/series/15585 Eric W. Hart James Sandefur (cid:129) Editors Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research 123 Editors EricW. Hart James Sandefur Mathematics Department Department ofMathematics andStatistics Grand View University Georgetown University DesMoines, IA Washington, DC USA USA ISSN 2520-8322 ISSN 2520-8330 (electronic) ICME-13 Monographs ISBN978-3-319-70307-7 ISBN978-3-319-70308-4 (eBook) https://doi.org/10.1007/978-3-319-70308-4 LibraryofCongressControlNumber:2017957205 ©SpringerInternationalPublishingAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface “Discrete mathematics is the math of our time.” So declared the immediate past president of the National Council of Teachers of Mathematics, John Dossey, in 1991 (as referenced in the first chapter, “Discrete Mathematics is Essential Mathematics in a 21st Century School Curriculum,” of this volume). Nearly 30 years later that statement is still true, although the news has not yet fully reached school mathematics curricula. Nevertheless, much valuable work has been done, and continues tobedone.This volume reports onsome of that work. It providesa glimpseofthestateoftheartinlearningandteachingdiscretemathematicsaround the world, and it makes the case once again that discrete mathematics is indeed mathematics for our time, even more so today in our digital age, and it should be included in the core curricula of all countries for all students. The chapters in this book are extended versions of papers presented at the thirteenth International Congress on Mathematical Education (ICME-13), held in Hamburg, Germany, in July 2016, as part of Topic Study Group 17 on discrete mathematics. The chapters are organized into six sections. The first section examinescurrentthinkingaboutdiscretemathematicsintheschoolcurriculum.The remaining sections focus on core discrete mathematics content and practices for school mathematics—combinatorics and combinatorial reasoning, recursion and recursive thinking, networks and graphs, fair decision-making and game theory, and, finally, logic and proof. A brief description of the chapters in each section follows. Part I: The Landscape of Discrete Mathematics in the School Curriculum Hart and Martin’s chapter launches the book. It identifies five essential discrete mathematics problem types that should be included in robust twenty-first century schoolcurricula.Besidesdiscussingtheseproblemtypesinsomedetail—including what they are, what mathematics is involved, and why these five, classroom v vi Preface examples are given that make the discussion immediately practical and imple- mentable. This chapter also serves as an introduction to the entire volume. Rosenstein looks backward and ahead at the school discrete mathematics movement in the United States. He describes the opportunity that discrete mathe- maticsprovidesforsupportingreasoning,problemsolving,andsystematicthinking in the school mathematics curriculum and illustrates this opportunity with a set of problems that begin “Find all ….” He analyzes the current lack of discrete math- ematicsinprimaryandsecondaryeducationintheU.S.,despiteastrongbeginning inthe1980sand1990s.Heprovidessomerationaleforwhythishashappened,why it is a counterproductive development, and then looks optimistically to a future in which discrete mathematics will play amore prevalent role inschool curricula. He includes specific recommendations and examples. InDiscreteMathematicsinLowerSchoolGrades?SituationandPossibilitiesin Italy,GaioandDiPaolafirstdescriberesultsofasurveytheyhavetakenover150 Italian mathematics teachers at all school levels. These results indicate that, espe- cially at the lower grades, teachers have minimal knowledge of discrete mathe- matics topics such as cryptography, algorithms, and graph theory. On the other hand, teachers indicate a willingness and desire to learn about these topics and to teach them in their classes. Much of this desire is based on wanting students to engage actively in mathematical problem solving. Later, the authors describe an ongoing project resulting from their earlier survey in which they are working with teachers to design and implement discrete mathematics activities involving binary numbers, algorithms, and cryptography in grades 3 through 8. InDiscreteMathematicsandtheAffectiveDimensionofMathematicalLearning and Engagement, Goldin addresses the importance of student attitudes, beliefs, emotions, and motivations when learning mathematics. The author argues that for many students, negative attitudes and emotions toward the traditionally taught curriculum can hinder their learning of mathematics. He goes on to describe some possible positive affordances arising by teaching discrete mathematics topics through problem solving contexts set in familiar experience. This can lead to exploration,enhancedengagement,andpersonalsatisfaction.Thechapterendswith acallformoreresearchontheaffectiveandconativedimensionsofthelearningand teaching of discrete mathematics. Part II: Combinatorics and Combinatorial Reasoning Combinatorics might be considered the mathematical art of counting. Combinatorialreasoningistheskillofreasoningaboutthesizeofsets,theprocess of counting, or the combinatorial setting to answer the question, How many? Combinatoricsisakeypartofdiscretemathematicsandcombinatorialreasoningis a powerful mode of reasoning, a mathematical habit of mind, that is specifically developed in the learning and teaching of discrete mathematics. Preface vii This section of the book begins with the chapter by Coenen, Hof, and Verhoef, Combinatorial Reasoning to Solve Problems, which studies the mathematical thinking of students, aged 14–16, as they try to solve combinatorial problems involving combinations and permutations, such as how many dishes of ice cream canwehaveusingthreeflavors,andhowmanyicecreamtowersonaconecanwe have. One result of their research is observing that students often start problem solving at the highest level of attention, which often leads to more mistakes and incorrectanswersthenforstudentswhostartatalowerlevel.Fromtheresearch,the authors observe that guidance from teachers is important. In fact, the chapter suggestshowstudents,guidedcorrectly,candeveloprelationalunderstandingusing combinatorial reasoning, which can lead to a better understanding of the problems and more confidence in their solutions. In Höveler’s chapter, Children’s Combinatorial Counting Strategies and their RelationshiptoMathematicalCountingPrinciples,theresultsofaqualitativestudy are presented, the main goals of which were to identify how children solve com- binatorialcountingproblemsandtogaininsightsintotherelationshipbetweentheir strategies and conventional mathematical counting principles. Counting strategies identified and discussed include multiplicative, additive, and compensation strate- gies, in addition to recursive strategies and comparing structural features of prob- lems.Thesestrategiesareexaminedwithrespecttotheirconceptualandoperational differences and similarities to combinatorial counting principles, including the multiplication principle, the principle of inclusion/exclusion, and the so-called shepherd’s principle. The chapter by Lockwood and Reed, Reinforcing Mathematical Concepts and Developing Mathematical Practices through Combinatorial Activity, focuses on a rich combinatorial task involving counting passwords. The authors provide examples of affordances that undergraduate students gained by engaging with the task. They highlight two kinds of affordances—those that strengthened under- standingaboutfundamentalcombinatorialideas,andthosethatfosteredmeaningful mathematical practices. They consider pedagogical implications and, in particular, maintain that combinatorics is an area of mathematics that offers students chances toengagewithaccessibleyetcomplexmathematicalideasandtodevelopimportant mathematical practices. They present examples of sophisticated student work that theyhopewillcontributetoanoverallgoalofelevatingthestatusofcombinatorics specifically, and discrete mathematics more broadly, in the school and under- graduate curriculum. The chapter by Vancso, et al. summarizes the ideas and background of a combinatorics research and teaching project, including historical reforms in the schoolcurriculumin1978inHungaryandT.Varga’swork.Thereaftertheydiscuss the main elements of their current project: a pretest and developed teaching materials, including student worksheets with rich problems and some tools for teaching combinatorics such as Poliuniversum. In choosing the problems for the worksheetsthey wereledbytworesearchquestions:(a)howstudentshandleopen tasks (which are presented in many of the combinatorial problems), and (b) how they use various manipulatives at different ages. viii Preface Part III: Recursion and Recursive Thinking Recursion involves describing a given step in a sequential process in terms of the previousstep(s).Suchadescriptionisoftencapturedinarecursiveformula(alsocalled adifferenceequation),whichmightbeinformal,suchasNEXT=NOW+2,orformal, suchasan=an−1+2,andisincontrasttoanexplicit(orclosed-form)formula,suchas a =2n.Recursionandrecursivethinkingarepowerfulmodelingandproblem-solving n strategies used throughout mathematics. They are particularly developed in the teachingandlearningofdiscretemathematics.Thetwochaptersinthissectiondiscuss thebenefitsofrecursionandrecursivethinkingintheclassroom,asseenparticularlyin thestudyofdifferenceequationsanddiscretedynamicalsystems. In Discrete Dynamical Systems: A Pathway for Students to Become Enchanted with Mathematics, Devaney points out that the traditional mathematics curriculum consists primarily offourth century, BC, geometry, eleventh century algebra, and possibly some seventeenth century calculus. He goes on to argue that to attract students to mathematics, they should have experiences in which they engage with some of the exciting areas of contemporary mathematics. Discrete mathematics offers a number of opportunities for engaging students in contemporary mathe- matics, as described throughout this monograph. In this chapter, the author describes a number of activities involving the modern field of discrete dynamical systems, particularly chaos and fractals. These activities, which have been suc- cessfullyusedwithstudentsforyears,leadtotheconstructionofsomestrangeand beautiful shapes. Combining recursion and iteration with traditional geometric topics, such as the geometry of transformations, leads to students developing an understanding of why these shapes arise. The activities involve exploration and creativity on the part of the students as they learn important mathematics. In How Recursion Supports Algebraic Understanding, Sandefur, Somers, and Danceproposetheintegrationofrecursivethinkingwithalgebraicthinking.Thechapter firstgivesanumberofsimplemodels,appropriateatavarietyofschoollevels,thatcan beapproachedusingbothstandardalgebraandrecursion.Thesemodels,buildingonthe ideasthatrepeatedadditionismultiplicationandrepeatedmultiplicationisexponen- tiation, lead to a more complete understanding of linear, quadratic, and exponential functions.Thefocusisoncovariationalthinking,particularlythedifferencesbetween constantandvariablechange,anunderstandingthatisatthecoreoflearningcalculus. Thechaptergivesavisionofhowthisintegrationcanbeachievedfromearlymiddle schoolthroughsecondaryschool,withinminimalchangeinthecurriculum. Part IV: Networks and Graphs The two chapters in this section show how teaching networks and graphs (also calledvertex-edgegraphs)cannotonlyhelpstudentslearnimportantmathematical content, but also foster mathematical thinking and give students the experience of Preface ix approachingproblemssimilarlytohowaresearchmathematicianmightapproacha problem. In Food Webs, Competition Graphs, and a 60-Year Old Unsolved Problem, Cozzens and Koirala define how a food web can be constructed by knowing the predator-prey relationships in a particular habitat. They then relate food webs to other types of graphs, most importantly competition graphs and interval graphs. Aftersomediscussionofthesedifferenttypesofgraphsandtheirinterrelationships, theauthorsproceedtodiscusshow,historically,thesegraphshavebeenusedtotry tounderstandrelationshipsbetweenspeciescompetingforthesameresources.This gives a nice example of how the use of contemporary mathematics, which is accessibleandrelevantforhigh schoolstudents,canleadtoabetter understanding of our world, ecological relationships in this case. InGraphTheoryinPrimary,MiddleandHighSchool,FerrarelloandMammana report on research they conducted on the introduction of graph theory in grades 3 through10inSicily. Theactivities centeraround theKönigsbergbridgesproblem, andmoregenerally,theideaofdeterminingwhenagraphhasanEulercycle.While thefocusissimilar,theleveloftheactivitiesisadjusteddependingongradelevel. The activities are described in some detail for the different grade levels and sum- maries of students’ responses to these activities are given. Part V: Fair Decision-Making and Game Theory Game theory is an area of mathematics dealing with situations of cooperation and conflict involving players, moves, strategies, and outcomes. Broadly viewed, it includes the mathematics offair decision-making as well as combinatorial games. Fair decision-making is the focus of Garfunkel’s chapter; Colipan and Rougetet consider combinatorial games. Garfunkel discusses a number of fairness models related to fair division and bankruptcy. Several models have a very long and colorful history. He emphasizes theroleofmathematicalmodelinginsolvingsuchfairnessandequityproblems.In addition to showing how accessible these discrete models can be, he attempts to show their intrinsic interest and the fact that they can and should be introduced in high school and even middle school mathematics curricula. InMathematicalResearchintheClassroomviaCombinatorialGames,Colipan describes the Chocolate Game, one of several Nim-type combinatorial games that can be used to give students an authentic research experience, similar to those of research mathematicians. In particular, the students consider questions that are mathematically easy to access, have a variety of strategies for going forward, and solutionstoonequestionbringoutnewquestions.Whilethesegamescouldbeused for a wide variety ofstudentlevels,thechapter includes research results when this approach was used with a group of 50 first-year college students. Rougetet’s chapter provides a rich history of the Nim game, a prototypical combinatorial game, and she considers the role of this game in the latest reform
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