Peter Grootenboer · Margaret Marshman Mathematics, Affect and Learning Middle School Students’ Beliefs and Attitudes About Mathematics Education Mathematics, Affect and Learning Peter Grootenboer Margaret Marshman (cid:129) Mathematics, Affect and Learning ’ Middle School Students Beliefs and Attitudes About Mathematics Education 123 PeterGrootenboer Margaret Marshman GriffithUniversity University of the SunshineCoast GoldCoastCampus, QLD SunshineCoast, QLD Australia Australia ISBN978-981-287-678-2 ISBN978-981-287-679-9 (eBook) DOI 10.1007/978-981-287-679-9 LibraryofCongressControlNumber:2015946086 SpringerSingaporeHeidelbergNewYorkDordrechtLondon ©SpringerScience+BusinessMediaSingapore2016 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. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 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 authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerScience+BusinessMediaSingaporePteLtd.ispartofSpringerScience+BusinessMedia (www.springer.com) PeterwouldliketodedicatethisbooktoJake, Danneke, and Talitha. Margaret would like to dedicate this book to Chaedyn and Cahlia—future middle years mathematics students. Preface For many years mathematics has been seen as a dull and boring subject that has littlerelevancetoreallife,andseeminglytheseviewsdevelopatschool.Thisbook focusesonthebeliefs,attitudes,valuesandemotionsofstudentsinYears5–8(aged 10–14 years) about mathematics and mathematics education. There has been con- siderable interest in the affective domain in mathematics education over the past two decades and many of the studies that have been with adult participants (often preserviceprimaryteachers).Whilethesestudieshavebeenusefulindescribingthe beliefs, attitudes and feelings of people about mathematics, and highlighting the poor image of the subject amongst much of the population, there appears to be a shortage of research actually undertaken with children. If indeed people are developingmanyoftheirresistantviewsaboutmathematicsduringtheirschooling, thenitisimportanttoactuallyinvestigate students’affectiveviewsofmathematics while they are at school. Clearly, negative views of mathematics can detrimentally impact student’s capacity to learn mathematics well at school and to use it effec- tively throughout their life. Conversely, students who are positively disposed towards mathematics are more likely to engage in mathematical activities and participate in opportunities that have a mathematical dimension. Fundamentally, this book focuses on this topic—the development of affective views and responses towards mathematics andmathematics learning. Furthermore, itseemsthatstudentsdeveloptheirmorenegativeviewsofmathematicsduringthe middle school years (Years 5–8), and so here we concentrate on students in these critical Years. This book is built on some empirical studies including qualitative enquiries with children in Years 5 and 8, and a large scale (n = 1784) quantitative study with students from a range of schools across diverse communities in New Zealand. While these studies were undertaken in a particular part of the world, the issues are common and have been reported across Australasia, Europe and the Americas, and so the findings have broad relevance and application. The studies revealed a number of key findings including: (cid:129) a view that mathematics is about numbers; (cid:129) the most important aspect of mathematics is learning your times-tables; vii viii Preface (cid:129) students’ emotional responses to mathematics and learning mathematics were related to their achievement and confidence; (cid:129) in general, the affective responses of students diminished as they moved up through the Years; (cid:129) there is a relationship between mathematical performance and attitudes to mathematics and beliefs about mathematics; and, (cid:129) despitethewidespreadhighachievementofgirlsinmathematics,variousgroups of students expressed a view that “boys were better than girls” in mathematics. The students’ affective responses to mathematics—their beliefs, attitudes and feelings, were grounded in their experiences, and these experiences were confined primarilytotheclassroom.Giventhatstudentsarelearningtheiraffectiveresponses to mathematics in school mathematics lessons, and their views seem to be dimin- ishing as they progress through their schooling, there is a need for a more holistic view of mathematics education in research, theory and practice—to consider the development of mathematical identities. This perspective promotes the simulta- neous consideration of the mathematical development in knowledge, skills, beliefs and attitudes. Students with healthy mathematical identities usually achieve better results in mathematics courses, will engage more wholeheartedly in their mathe- maticallearning,aremorelikelytocontinuewiththeirstudiesinmathematics,and will more readily participate in mathematical experiences. A feature of this book is that it brings some substantial empirically-based understandings to the widely held perception that many students have negative views of mathematics, and these affective responses develop during the middle years of school. Specifically, the data for this book were collected with school students, and students who were actually engaged in learning mathematics in their crucialmiddleschoolyears.Therefore,thefindingsreportedanddiscussedhereare relevant for researchers and mathematics educators, policy makers and curriculum developers, and teachers and school leaders engaged in the teaching of mathematics. Acknowledgments First, wewant toexpressoursincerest thanks totheteachers,students andschools in Australia and New Zealand who participated in the research projects we have drawnonhere.Weappreciatethehonestyandinsightfulnessofyourresponsesand theyhaveallowedustounderstandmorefullymathematicseducationinthemiddle school years. Second,wewanttoexpressourthankstoourcolleagueswhoreaddraftversions of this book and provided thoughtful feedback and questions. This included Dr.KevinLarkinandDr.HarryKanansaatGriffithUniversity,Dr.ShaileighPage, and Associate Professor Peter Dunn and Dr. Rob McDougall at University of the SunshineCoast.Dr.AaronWiegandproducedsomeofthegraphs.Wealsowishto acknowledge the support of our professional community—MERGA, and our col- leagues who have helped us develop and refine our ideas and thoughts. PeterwouldspecificallyliketothankhiscolleaguesinthePedagogy,Education and Praxis research collaboration group. Although these colleagues aren’t specif- ically interested in mathematics education, they so provide a thoughtful academic community that stimulates theoretical and practical educational consideration. He alsowishestothankhiscolleaguesintheGriffithInstituteforEducationalResearch forpracticalandintellectualsupportovermanyyears.Finally,Peterwantstothank his family and friends for their generosity, love and support over the time of this researchandwriting.Specifically,Ange,Jake,DannekeandTillyathome;Willem, Beverley,Janine,MattandMichelleaswegrewup;and,Murray,PeterandNickon the golf course. Margaret would especially like to thank Prof. Chris Bigum for showing her the way in education research, with those many thought-provoking discussions, read- ingsandquestionsthatalwaysresultedinherhavingmorequestionsthananswers. She would also like to thank her family Barry, Justin, Cara, Caitlin and Ryan for their love, support and encouragement whilst researching and writing. ix Contents 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The Importance of Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Researching and Understanding Affect in Mathematics Education . . . . 3 Learning in the Middle Years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Significance of the Teacher. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Outline of the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 The Affective Domain, Mathematics, and Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 The Affective Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Changing Beliefs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Beliefs About Mathematics and Mathematics Education . . . . . . . . . . . 17 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Attitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Attitudes to Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Emotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 The Affective Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Affect and Mathematics Education. . . . . . . . . . . . . . . . . . . . . . . . 22 Mathematical Affect and Learning Mathematics. . . . . . . . . . . . . . . 23 Mathematical Affect and Mathematical Achievement . . . . . . . . . . . . . 25 Mathematical Affect and Future Mathematical Engagement. . . . . . . . . 26 Mathematics Education as Developing Mathematical Identities . . . . . . 27 Overview and Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Investigating Students’ Ideas About Mathematics and Mathematics Education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Researching the Affective Domain in Mathematics. . . . . . . . . . . . . . . 36 xi xii Contents Kids Talk About Maths Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Data Collection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Key Themes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Kids Ideas About Maths Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 The Instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 The Participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 The Affective Subscales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Secondary Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Key Findings and Themes from Across the Studies . . . . . . . . . . . . . . 47 Mathematics Is About Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . 48 Times-Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Other General Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4 Students’ Beliefs and Attitudes About Mathematics and Learning Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 The Importance of Mathematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Beliefs About the Nature of Mathematics . . . . . . . . . . . . . . . . . . . . . 58 Utilitarian Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Traditional Beliefs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Beliefs About Learning Mathematics . . . . . . . . . . . . . . . . . . . . . . 64 Attitudes Toward Mathematics and Learning Mathematics . . . . . . . . . 67 Mathematics Anxiety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5 Differences in the Affective Responses of Various Groups . . . . . . . 75 Gender. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Cultural Identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Mathematical Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Socio-Economic Status (SES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Type of School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6 Changes in Affective Responses to Mathematics Through the Middle School Years. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Sliding Affective Responses to Mathematics . . . . . . . . . . . . . . . . . . . 92 Positive Attitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Utilitarian Beliefs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Traditional Beliefs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98