Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers

465 Pages·2018·4.75 MB·English

by  Nicholas H. Wasserman

Research in Mathematics Education
Series Editors: Jinfa Cai · James A. Middleton
Nicholas H. Wasserman Editor
Connecting
Abstract Algebra
to Secondary
Mathematics,
for Secondary
Mathematics
Teachers
Research in Mathematics Education
Serieseditors
JinfaCai,Newark,DE,USA
JamesA.Middleton,Tempe,AZ,USA
Thisseriesisdesignedtoproducethematicvolumes,allowingresearcherstoaccess
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isparticularlygearedtowardhighlightingtheworkofpromisinggraduatestudents
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andmathematicseducationleaders—peoplewhoneedthehighestqualityresearch,
methodological rigor, and potentially transformative implications ready at hand to
helpthemmakedecisionsregardingtheimprovementofteaching,learning,policy,
andpractice.Withthisvision,ourmissionofthisbookseriesis:
1. To support the sharing of critical research findings among members of the
mathematicseducationcommunity;
2. Tosupportgraduatestudentsandjuniorfacultyandinductthemintotheresearch
communitybypairingthemwithseniorfacultyintheproductionofthehighest
qualitypeer-reviewedresearchpapers;and
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Moreinformationaboutthisseriesathttp://www.springer.com/series/13030
Nicholas H. Wasserman
Editor
Connecting Abstract Algebra
to Secondary Mathematics,
for Secondary Mathematics
Teachers
123
Editor
NicholasH.Wasserman
DepartmentofMathematics,ScienceandTechnology
TeachersCollege,ColumbiaUniversity
NewYork,NY,USA
ISSN2570-4729 ISSN2570-4737 (electronic)
ResearchinMathematicsEducation
ISBN978-3-319-99213-6 ISBN978-3-319-99214-3 (eBook)
https://doi.org/10.1007/978-3-319-99214-3
LibraryofCongressControlNumber:2018962925
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Foreword
“Ijustwanttoteachhighschoolmathematics,whyshouldItakeabstractalgebra?”
“Why would taking abstract algebra help me become an effective high school
mathematics teacher?” For those of us with experience in secondary mathematics
teachereducation,thesearenotunfamiliarquestions.Thisvolumecanhelpanswer
these questions. Indeed, secondary teachers’ mathematical knowledge, and, more
specifically, their lack of mathematical knowledge, is a common but naïvely cited
reason for the myriad problems we encounter as we try to improve mathematics
teaching and learning in high schools. Although merely taking more advanced
courses has long been shown to be ineffective at improving student learning
outcomes, this volume focuses on particularly useful connections which can be
made from the study of abstract algebra—a mathematics course several years
beyondwhathighschoolteachersareexpectedtoteach—andteachers’knowledge
ofanddisciplinarypracticesrelatedtothemathematicsrequiredintheirhighschool
classes.
Someideasinthisbookaretheoreticalandspeculative.Thehistoricaldevelop-
ment of mathematical ideas, for example, is illuminative for those of us interested
in the epistemology of mathematical understanding. These ideas highlight what
couldbepossibleifteachers’collegiateandprofessionaleducationwerefocusedon
the “higher standpoint.” Teachers may develop new tasks, stories, and discussions
that introduce students to some of these major leaps of understanding and their
associated practices, but only if they understand them and their application to the
contenttheyareexpectedtoteach.
Themainbodyofthebook,therefore,isempirical,drawingupontheexperience
and careful research and development conducted by the authors. One of the key
innovations of this book is the attention to not just how abstract algebra informs
high school content (though it does this, in our estimation, better than any other
volume on the subject) but how the situations and dilemmas that arise from
teaching high school mathematics can provide a meaningful structure for learning
advanced mathematics. Moreover, this mutual impact—making connections from
advancedmathematicstohighschoolmathandfromhighschoolmathtoadvanced
v
vi Foreword
mathematics—isnotmerelyfocusedoncontentorhistorybutonthewaysinwhich
people have grappled with algebraic ideas that connect learning and instruction
throughthepracticesandhabitsofmindthatmarkourfieldasunique.
Mathematics and its teaching is unique in the world of academic subjects,
and it is often only in grappling with advanced mathematics—stretching our own
understanding of content and mathematical practices—that we can see beyond
the common conception of mathematics as a subject that exists primarily to serve
engineeringandtheappliedsciences.Perhapsbyadvancingtheideaspromotedin
thisbookwecanhelphighschoolstudentsseethisaswell.
Because of this duality and because of the rigorous theoretical and empirical
standpointtakenbytheauthors,theexamplesandinstructionalimplicationsinthis
volumeareofimmensepracticalutilityforthoseofustaskedwiththepreparation
andprofessionallearningexperiencesofsecondarymathematicsteachers.Assuch,
thisbookepitomizesoneoftheprimaryrationalesthatwe,asco-editors,provided
when we proposed this series: to increase the utility of mathematics education
research for those who would impact practice at the cutting edge. We wish to
expressourgreatappreciationtotheauthorsofthechaptersinthisvolumeand,of
course, the editor, Nick Wasserman, for creating such a melding of research and
practice.
DepartmentofMathematics JinfaCai
UniversityofDelaware
Newark,DE,USA
IraA.FultonSchoolsofEngineering JamesA.Middleton
SchoolforEngineeringofMatter,
Transport&Energy
ArizonaStateUniversity
Tempe,AZ,USA
Acknowledgments
The work that goes into the creation of a volume is both in conceptualization and
realization, and I would like to acknowledge people who played a role in each.
First,Iwouldliketothankthemanycolleagues(toomanytolist,althoughseveral
of whom are authors of chapters) who have engaged with me in thinking about
advanced mathematics and secondary teacher education and who have challenged
meandpushedmetothinkmoredeeplyandmorecriticallyinthisarea.Moreover,
this volume would not exist without the initial encouragement from Rina Zazkis
to bring together some of the current work around, and the existing contributions
about, abstract algebra and secondary teaching. To all of these colleagues, thank
you for your role in helping conceptualize this volume and in shaping many of
the ideas in it. Second, I am grateful for the work of all the authors, commentary
authors, reviewers, series editors, copyeditors, etc., for helping bring this volume
intoexistence.Inthisregard,inadditiontotheauthorsyouseewithinthevolume,
IwouldliketothankJonathanBostic,ShawnBroderick,JinfaCai,PaulDawkins,
BenDickman,TimFukawa-Connelly, StevenGreenstein,WillMcGuffey,Melissa
Mills, Teo Paoletti, Melissa Renga, Jason Samuels, Julianna Stockton, Megan
Wawro,AnnWheeler,SeanYee,andBillZahner.
vii
Contents
1 ExploringAdvancedMathematicsCoursesandContent
forSecondaryMathematicsTeachers.................................... 1
NicholasH.Wasserman
PartI Exploring Students’ Learning of Abstract Algebra
ConceptsCloselyConnectedtoSecondaryMathematics
2 ConnectingtheGroupTheoryConceptAssessmenttoCore
ConceptsattheSecondaryLevel ......................................... 19
KathleenMelhuishandJoshuaFagan
3 Monster-Barring as a Catalyst for Bridging Secondary
AlgebratoAbstractAlgebra .............................................. 47
JohnPaulCook
4 WhatKindofOpportunitiesDoAbstractAlgebraCourses
ProvideforStrengtheningFutureTeachers’Mathematical
KnowledgeforTeaching?.................................................. 71
SeanLarsen,ErinGlover,AnnaMarieBergman,andJohnCaughman
PartII ExploringMathematicalConnectionsBetweenAbstract
AlgebraandSecondaryMathematics
5 ExcavatingSchoolMathematics .......................................... 87
WilliamMcCallum
6 Foreshadowing Ideas in Abstract Algebra: Reflections
from the Secondary School Mathematics Curriculum
ImprovementStudy ........................................................ 103
J.PhilipSmith
7 Understanding School Mathematics in Terms of Linear
MeasureandDiscreteRealAdditiveGroups............................ 125
HymanBass
ix
x Contents
8 Abstract Algebra and Secondary School Mathematics
ConnectionsasDiscussedbyMathematiciansandMathematics
Educators.................................................................... 149
AshleyL.Suominen
9 MakingMathematicalConnectionsBetweenAbstractAlgebra
and Secondary Mathematics Explicit: Implications for
Curriculum,Research,andFacultyProfessionalDevelopment....... 175
JamesA. MendozaÁlvarezandDianaWhite
PartIII ExploringApproachestoSecondaryTeacherEducation:
EngaginginAbstractAlgebrainRelationtoDeveloping
DisciplinaryPractices
10 MakingConnectionsfromtheSecondaryClassroomtothe
AbstractAlgebraCourse:AMathematicalActivity
Approach .................................................................... 189
RoseMaryZbiekandM.KathleenHeid
11 LearningMathematicalPracticestoConnectAbstractAlgebra
toHighSchoolAlgebra .................................................... 211
ErinE.Baldinger
12 FromEquationstoStructures:ModesofRelevanceofAbstract
Algebra to School Mathematics as Viewed by Teacher
EducatorsandTeachers ................................................... 241
JosephineShamash,MaritaBarabash,andRuhamaEven
13 UsingGeometricHabitsofMindtoConnectGeometryfrom
aTransformationPerspectivetoGraphTransformationsand
AbstractAlgebra ........................................................... 263
YvonneLaiandAllanDonsig
14 DevelopingaStructuralPerspectiveandItsRoleinConnecting
SchoolAlgebraandAbstractAlgebra:AFactorizationExample .... 291
YounheeLeeandM.KathleenHeid
15 BuildingaCoherentResearchProgramthatLinksAbstract
AlgebratoSecondaryMathematicsPedagogyviaDisciplinary
Practices ..................................................................... 319
JamesCummings,EliseLockwood,andKeithWeber
PartIV ExploringApproachestoSecondaryTeacherEducation:
EngaginginAbstractAlgebrainRelationtoPedagogical
ProblemsinTeachingSecondaryMathematics
16 ExploringanInstructionalModelforDesigningModulesfor
SecondaryMathematicsTeachersinanAbstractAlgebraCourse ... 335
NicholasH.WassermanandPatrickGalarza