Table Of ContentSepideh 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
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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.
