And the Rest is Just Algebra PDF
Preview And the Rest is Just Algebra
Sepideh Stewart E ditor And the Rest is Just Algebra And the Rest is Just Algebra Praise for “And the Rest Is Just Algebra” Middleschool,collegealgebra,calculus,andlinearalgebrateacherscanallbenefit fromthisbook.Becomingproficientwithalgebraisacomplextask—muchmoreso than it appears to those who were successful the first time around. Furthermore, according to neuroscience studies, manipulating symbolic expressions requires considerable cognitive effort even for those who are proficient. In addition, “met-befores” in the form of prior arithmetic thinking can thwart algebraic think- ing.Allthis,andmore,isconsideredinareadablewayinthisbook. AnnieSelden Sepideh Stewart Editor And the Rest is Just Algebra Editor SepidehStewart DepartmentofMathematics UniversityofOklahoma Norman,OK USA ISBN978-3-319-45052-0 ISBN978-3-319-45053-7 (eBook) DOI10.1007/978-3-319-45053-7 LibraryofCongressControlNumber:2016953106 ©SpringerInternationalPublishingSwitzerland2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof 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 dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthis 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, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Foreword Themathematicseducationcommunityhashadakeeninterestinstudents’learning of algebra, especially since the “Algebra for All” movement began in the 1990s. The great bulk of research on students’ algebra, however, focuses on students’ development of algebraic thinking in the early grades, on obstacles students must overcometoreason algebraically,andonknowledgerequiredofteacherstoteach algebrawell.Thatis,thevastmajorityofresearchonstudents’algebrahasfocused on learning and teaching algebra. Very little research has focused on the various consequences for students’ later mathematical learning of the algebra that they actuallylearn.Thisbookpicksupthatfocus.Itaddressestheconsequencesofthe algebra students learn in school for their mathematical learning in college. It also addresses a problem created for students by college instructors’ attitude that, in solving mathematical problems or proving theorems, “We’ve done the hard, stra- tegicpart.Therestisjustalgebra.”Itturnsoutthat“therest,”forstudents,isnot just algebra. It too often is an abyss they fall into with great apprehension and anxiety,anabyssthatdistractsthemfromstrategyandmeaningandobstructstheir insightintolargerthemes. Actually, this book does more than trace consequences of students’ difficulties withschoolalgebra.Chaptersinitoffernewinsightsinto (cid:129) Sourcesofstudents’difficultiesinalgebra(e.g.,impoverishedunderstandingsof fractionandproportion) (cid:129) Theinterplaybetweenaffectandmathematicalunderstanding (cid:129) Theroleofreflectioninstudents’successfulalgebralearning (cid:129) Alternativeconceptionsofalgebra(e.g.,emphasisonfunctionsandmodeling) (cid:129) Thecentralityofgeneralizingandparticularizinginalgebraicthinking (cid:129) Critical ways of thinking that curricula and instruction failto foster (e.g., vari- ablesvaryandexpressionshavenumericalvalues) (cid:129) Uses of computer algebra systems to help students develop symbol sense and structuresensewhile,ironically,removinganyneedthattheyengagedirectlyin algebraicmanipulations v vi Foreword (cid:129) Newwaystothinkaboutlinearalgebrathatbothdeepenandbroadenstudents’ schoolalgebra. Putanotherway,thisbookcontainsacollectionofchaptersthatputpastresearch onstudents’algebrainanewlight.Italsooffersnewwaystothinkaboutaddressing students’ difficulties in school algebra while at the same time offering ways to envisionhowwemightsupportstudents’meaningfulusesofalgebrabeyondschool mathematics. Tempe,AZ PatrickW.Thompson June2016 Introduction In solving mathematics problems in college level, very often one reaches a point where certain algebraic manipulation is a required and unavoidable part of the procedure. Once a mathematician reaches this point the rest is trivial, straightfor- ward,andinsomesenseapleasantcompletiontoalongprocess.Namely,thereal mathematics part is taken care of “and the rest is just algebra,” an expression commonly used by many mathematics instructors. On the contrary, many college students undergo an entirely different experience. These students often halfheart- edly apply the more advanced theories and make some progress in a problem- solvingsituation;however,thealgebraportionswhichareinitiallyhiddenormixed in with other contexts create an enormous obstacle. The unresolved high school algebraknowledgeisanalogoustoatornadowrappedinrain,appearinginsomany courseswhereoutwardlyseemfarremovedfromhighschoolalgebra.Incollegeas thecomplexityofmathematicalideasincreasesrapidly,theunresolvedhighschool algebra problems mount up progressively and continue to create further distress. While assessing students’ work, it is often difficult to unravel their thought pro- cesses and the convoluted algebraic errors which are challenging to accurately categorizeorjustify. Many college instructors are facing this dilemma every day. Students who seemingly follow more complex mathematical concepts are unable to proceed as problems, for example involving fractions, will soon let them down. In college vii viii Introduction level, students who cannot perform what is known as “basic algebraic manipula- tions,”ordonotpossessadequate“assumedalgebraknowledge”readilyavailable, facesignificantdisadvantages.Thiscohortofstudentsareoftennotabletofollow the instructors’ problem-solving steps and easily get lost if any steps are skipped. Theyareoftenunabletosuccessfullyreachadesiredsolutionontheirown,causing frustrations for themselves, their instructors, and consequently raising many con- cernsfortheinstitutionandthecountryasawhole. Thisbookisacoordinatedcollectionofchapterswrittenbyseveralexpertsinthe fieldthataddressesoneofthemostpersistentmathematicspedagogychallengesof thiscentury.Theauthors,whohavecriticallyexaminedstudents’difficultiesfrom theirareasofresearch,emphasizethatthesedifficultiesaremorecomplexthanjust forgottenrulesandofferstrategicapproachesthatholdpromiseofgreatersuccess formorecollegestudents.Theirresearchanddiscussionwillraiseawarenessonthe complexity and challenges facing the mathematics community. Mathematics instructors who are frustrated with their students’ lack of skills and knowledge willfindthisvolumehelpful,astheauthorsconfrontthequestionofwhystudents have difficulties with algebra and reveal how to improve their long-term under- standingandsuccess. The first part of the book brings issues regarding the current state of students’ inadequacies and fluency of algebra in the US colleges and is divided into two chapters. In the first chapter, Stewart and Reeder set the scene by showing how the unresolved high school algebra misconceptions and shortcomings create major complications in college mathematics courses. The examples that are illustrated inthischapterarethetipoftheiceberg,showingthetypesthatmostinstructorswill frequently come across in assessing students’ work. The consequences of lacking solid algebra in college level are devastating and deserve to be addressed appropriately. The second chapter by McGowen shows how the problematic nature of prior knowledge hinders students’ success in college-level mathematics. She cites the needtoimprovetheeffectivenessofteachingandcallsforteacherstobeawareof students’problematicmet-befores,providingguidanceaccordingly,inordertohelp studentsdevelopdeeperunderstandingofmathematicsandpromotemathematical thinking. Thesecondpartofthebook,devotedtoalgebrainabroadercontext,isdivided intothreechapters. Inthefirstchapter,Talloffersacombinedframeworkformathematicsingeneral and algebra in particular and contemplates why some students find algebra plea- surable whereas others find it a source of anxiety. In his view, many students are affected by problematic aspects that have accumulated over many years and become more difficult to address as the ideas become deeply ingrained. Tall suggests a significant re-think in how we view the development of mathematical thinking that promotes flexible thinking on the one hand and impedes long-term learningontheother. Introduction ix The second chapter presented by Booth, McGinn, Barbieri, and Young is concerned about students’ common misconceptions. The authors believe that tra- ditional instruction will not remedy the problem. Drawing from a wealth of mathematics education and educational psychology literature, the authors offer a numberofinterventionstoaddressthesemisconceptions.Furthermore,theydiscuss waysofpreventingthemfromdeveloping. ThethirdchapterbyReederhighlightsthesignificanceofmathematicsteachers’ content and pedagogical knowledge. In her views, to prepare successful students whoareabletocontributetotheglobaleconomy,wemustinsistthatthemastryof basicskillsisnotsuffecient.Inherviews,theprocessofpreparingstudentswellis challengingandrequiresaholisticapproach,butcanbemet. The third part of the book, devoted to positive approaches to the teaching of algebra,isdividedintotwochapters. Drawing on years of experience working with many students from all ages, Mason believes that once the learner appreciates where the algebraic expressions comefrom,manipulating them isastraightforwardtask.Thischapteroffers well- thoughtoutpedagogicalstrategies,didactictactics,andspecificallydesignedtasks andofferswaysforwardonhowtosucceedinlearningalgebra. FeyandSmithsuggestaboldcurriculumchange,centeringonfunctions.They assert that the current US high school curriculum has no room for more applied mathematics. Topics such as probability, statistics, modern discrete mathematics, andmathematicalmodelingmustbeincludedtoprepareourstudentstosolvereal life problems. In this chapter, the authors assess their proposed curriculum by carefullyconsideringthechallengesandrespondingtothem. Combining historical and current didactical ideas, Nataraj and Thomas reveal fresh approaches to the understanding of algebra. The authors discuss the various uses of letters and the concept of exponentiation and emphasize that in order to preparestudentsforlong-termunderstandingandsuccess,muchgroundworkneeds tobeestablishedinthemiddleandlowersecondaryyearsofschooling. Thefourthpartofthebookproposesfuturedevelopmentsandisdividedintotwo chapters. Drawingfromstudiesinmathematicseducationalneuroscience,Kieranreveals newfindingsofferingdifferentinsights intohowwethinkabouttheso-calledjust algebrapartofamathematicalproblem.Kieranencouragesthemathematicscom- munitytoreflectandquestiontheirtraditionalbeliefsthatassumealgebraicactiv- itiesarestraightforwardalgorithmicprocedures. ThesecondchapterpresentedbyThomasexamineshowparticulartasks,includ- ing some that integrate digital technology into student activity, could be used to re-think the algebra curriculum content with a view to motivating students and promotingversatilethinking.ThomasfindstheunderlyingprinciplesofFramework of Advanced Mathematical Thinking (FAMT) prove to be useful in school level mathematics. The last part of the book, dedicated to teaching higher algebra, is divided into twochapters.
The list of books you might like
