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Towards"antideficit"education"in"undergraduate"mathematics"education:"How"
deficit"perspectives"work"to"structure"inequality"and"what"can"be"done"about"it"
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Frederick"Peck"
University"of"Montana"
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frederick.peck@umontana.edu""
+1"406.243.4053"
@frederickpeck"
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Citation:""
Peck, F. A. (2020). Towards antideficit education in undergraduate mathematics
education: How deficit perspectives work to structure inequality and what can be
done about it. PRIMUS. Online first publication.
https://doi.org/10.1080/10511970.2020.1781721
! !
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Abstract:"A"deficit"perspective"is"a"propensity"to"locate"the"source"of"academic"
problems"in"deficiencies"within"students,"their"families,"their"communities,"or"their"
membership"in"social"categories"(such"as"race"and"gender)."While"the"deficit"
perspective"is"a"common"topic"of"discussion"in"K12"mathematics"education,"it"is"a"
much"rarer"topic"of"discussion"in"undergraduate"mathematics"education."I"argue"
that"undergraduate"mathematics"instructors"should"take"the"deficit"perspective"
seriously,"because"it"structures"educational"experiences"for"students"and"
instructors."In"this"survey"article,"I"provide"a"brief"review"of"the"literature"on"deficit"
perspectives"in"mathematics"education,"and"I"offer"practical"suggestions"for"
education"from"an"antideficit"perspective."
"
Keywords:"Deficit"perspective,"antideficit,"assets,"assetbased"education"
"
"!
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Towards"antideficit"education"in"undergraduate"mathematics"education:"How"
deficit"perspectives"work"to"structure"inequality"and"what"can"be"done"about"it"
1. INTRODUCTION!
For"the"past"four"decades"policymakers"and"educators"in"mathematics"have"been"
preoccupied"with"socalled"“gaps”"between"student"groups."For"example,"on"many"
largescale"standardized"tests"of"mathematics,"the"mean"scores"for"White"and"Asian"
students"are"higher"than"the"mean"scores"for"Black"students,"Latinx"students,"and"
students"from"other"systemically"oppressed"groups
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."This"has"been"branded"the"
“achievement"gap”"(Reardon"&"Robinson,"2007)."Within"undergraduate"
mathematics,"there"is"also"a"“participation"gap”"by"gender"and"by"race/ethnicity."
White"males"are"over"represented"in"collegelevel"mathematics,"while"other"groups"
are"underrepresented"(C."Rodriguez"et"al.,"2012;"Smith,"2011)."Similar"“gaps”"exist"
by"social"class,"with"upper"and"middleclass"students"having"higher"mean"scores"on"
standardized"tests"and"enjoying"overrepresentation"in"collegelevel"mathematics
2
.""
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"I"use"this"term"to"refer"to"people"who"have"systemically"been"denied"access"to"valued"cultural"
resources"based"on"sociallyconstructed"category"systems."I"use"the"term"“oppressed”"instead"of"the"
more"common"term"“underrepresented”"because"I"view"underrepresentation"as"an"outcome"of"
oppression."Using"the"term"oppression"foregrounds"the"cause,"not"the"effect,"and"makes"salient"the"
fact"that"underrepresentation"is"neither"natural"not"neutral."That"said,"I"realize"that"the"term"
“oppressed”"can"conjure"images"of"people"without"hope"or"agency."As"I"describe"later"in"this"paper,"
that"is"an"inaccurate"perception"of"people"in"oppressed"groups,"and"it"is"not"my"intention"to"
perpetuate"it."
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"Focusing"on"“gaps”"is"problematic"for"many"reasons"(R."Gutiérrez,"2008;"Martin,"2009)."Chief"among"
these"reasons"is"because"such"a"focus"promotes"a"deficit"perspective,"the"topic"of"this"review."Thus"I"
begin"the"review"with"a"discussion"of"“gaps”"not"to"reify"the"term,"but"rather"because"in"recent"
history"the"discourse"on"“gaps”"has"been"a"large"driver"of"deficit"perspectives."Due"to"this"history,"I"
use"quote"marks"to"mark"“gaps”"as"a"problematic"term.`"
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One"response"to"these"“gaps”"is"to"attribute"them"to"some"sort"of"deficiency"in"
oppressed"groups."This"tendency"has"come"to"be"known"as"a"deficit'perspective."
Deficit"perspectives"are"pervasive"in"STEM"(Science,"Technology,"Engineering"and"
Mathematics)"education"(Castro,"2014),"and"they"have"a"profound"effect"on"
students’"educational"experiences"and"opportunities"(Adiredja"&"AndrewsLarson,"
2017;"Adiredja"&"Louie,"2020).""
Discourse"about"deficit"perspectives"is"commonplace"in"the"K12"educational"
literature"and"there"is"an"emerging"body"of"powerful"work"that"discusses"the"deficit"
perspective"in"undergraduate"STEM"education"(Adiredja,"2019b,"2019a,"2020;"
Adiredja"&"AndrewsLarson,"2017;"Adiredja,"BélangerRioux,"&"Zandieh,"2020;"
Castro,"2012,"2014;"Hagman,"2019;"Leyva,"2016)."In"this"article,"I"aim"to"contribute"
to"this"literature"by"providing"a"practitionerfocused"survey"of"the"literature"related"
to"the"deficit"perspective,"and"practical"suggestions"for"antideficit"education."The"
paper"is"structured"as"follows."In"the"next"section"I"define"the"deficit"perspective,"
provide"a"brief"history,"and"give"examples"from"the"STEM"education"literature."I"
then"discuss"why"it’s"important"that"we"understand"deficit"perspectives,"showing"
how"they"work"to"structure"experiences"for"students"and"instructors."Finally,"I"
describe"alternative"models"for"education"from"an"antideficit"or"asset"perspective."
2. HISTORY!AND!EXAMPLES!OF!THE!DEFICIT!
PERSPECTIVE!
Box"1"provides"a"definition"of"the"deficit"perspective"in"educational"contexts.""
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Definition:
The deficit perspective is a propensity to locate the source of academic problems in
deficiencies within students, their families, their communities, or their membership in social
categories (such as race and gender)."
Box"1."Definition"of"the"deficit"perspective"
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While"contemporary"uses"of"the"deficit"perspective"are"often"associated"with"
attempts"to"explain"achievement"and"participation"gaps,"the"deficit"perspective"has"a"
muchlonger"history,"stretching"back"to"the"beginning"of"the"socalled"
enlightenment"period"and"the"social"construction"of"race"(Valencia,"2010)."Whereas"
historically,"deficit"perspectives"were"levied"on"racial"groups,"contemporary"deficit"
perspectives"in"education"are"directed"at"many"levels"including"individual"students,"
families,"communities,"and"social"groups."Deficit"perspectives"come"from"a"variety"of"
sources,"including"instructors,"administrators,"and"policy"documents"(Shields,"
Bishop,"&"Mazawi,"2005)."Box"2"shows"four"examples"of"contemporary"deficit"
perspectives"in"STEM."In"Example"A,"a"university"administrator"exhibits"a"deficit"
perspective"toward"individual"students."In"Examples"B"and"C,"instructors"
demonstrate"deficit"perspectives"towards"students"(Example"B)"and"families"
(Example"C)."In"Example"D,"a"prominent"policy"document"takes"a"deficit"perspective"
towards"families"and"social"groups"by"suggesting"that"they"do"not"have"a"history"of"
engagement"in"mathematical"practices.""
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Example A (from administrator towards students):
Postsecondary Administrator: “…they [the students in our program] are not prepared.” (Castro,
2012, p. 3, p. 3)
Example B (from instructor towards students):
Instructor (describing their perception of why students did not enter school ready to learn): “One
of the problems with the kids we’re putting in [the 2year algebra course] is they don’t have the
logic component.” (Horn, 2007, p. 62, p. 62)
Example C (from instructor towards families):
Instructor: “Today a lot of these kids are from broken homes. They have parents who are
criminals.” (García & Guerra, 2004, p. 159, p. 159)
Example D (from policy document towards families and social groups):
Policy document: “In the United States, mathematics is primarily part of upper and middleclass
male culture. Except for shopkeeper arithmetic of a bygone age taught in the elementary school,
few parts of mathematics are embedded in the family or cultural traditions of the many large
‘developing countries’ that make up the American mosaic.” (National Research Council, 1989, p.
20) (as cited in Martin (2009, p. 309). Italics added by Martin)"
Box"2."Four"examples"of"the"deficit"perspective.""
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A"preferable"perspective"is"to"attend"to"systemic"oppression."For"example,"
some"scholars"locate"oppression"in"the"ways"that"school"systems"have"historically"
and"contemporaneously"operated"to"deny"access"to"opportunities"based"on"a"
person’s"skin"color,"ethnic"background,"gender,"or"social"class."These"scholars"
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reframe"the"achievement"gap"as"an"“opportunity"gap”"(Carter"&"Welner,"2013)"or"
“education"debt”"(LadsonBillings,"2006).""
One"might"question"the"utility"of"this"reframing."Is"it"just"an"issue"of"semantics,"and"if"
so,"what"difference"does"it"make?"I"address"this"question"in"the"next"section."
3. EFFECTS!OF!THE!DEFICIT!PERSPECTIVE!
The"deficit"perspective"is"more"than"just"a"semantic"turn"of"phrase."Rather,"the"
deficit"perspective"works"to"structure"students’"and"educators’"experiences"in"STEM"
education."In"particular,"the"deficit"perspective:"(a)"limits"access"to"educational"
opportunities;"(b)"results"in"lowered"expectations"for"students;"(c)"limits"the"role"
that"instructors"can"play"in"a"student’s"education;"and"(d)"works"with"“culture"free”"
beliefs"about"STEM"curricula"to"prevent"powerful"forms"of"cognition"from"entering"
the"classroom."Perhaps"most"perniciously,"deficit"perspectives"keep"all"of"the"above"
hidden,"thus"(e)"preventing"critical"introspection;"and"(f)"perpetuating"oppression"
and"privilege."Below,"I"elaborate"on"each"of"the"above."
The'deficit'perspective'limits'educational'opportunities'and'results'in'lowered'
expectations'for'students"(Adiredja,"2019a,"2020;"Adiredja"et"al.,"2020;"Adiredja"&"
Zandieh,"2020;"Ford"&"Grantham,"2003;"Ford,"Harris,"Tyson,"&"Trotman,"2001;"
García"&"Guerra,"2004;"Horn,"2007;"Martin,"2006,"2009;"Shields"et"al.,"2005;"
Valencia,"2010)."It"is"welldocumented"that"members"of"traditionally"oppressed"
groups"are"overrepresented"in"remedial"courses"(Adelman,"2000;"Sparks"&"Malkus,"
2013),"and"that"the"sorting"of"students"into"such"courses"is"not"independent"of"
privilege"(DarlingHammond,"2010;"Oakes,"1990)."These"courses"are"often"focused"
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on"memorizing"facts"and"procedures"rather"than"engaging"students"in"cognitively"
demanding"mathematical"activity"(Larnell,"2016;"Oakes,"1990)."The"result"is"that"
students’"individual"experiences"with"STEM"are"racialized"(Larnell,"2016;"Martin,"
2006;"Valenzuela,"1999)"and"gendered"(Burton,"1995),"and"the"macrolevel"result"is"
the"disproportionate"access"to"opportunity"discussed"above.""
The'deficit'perspective'limits'the'role'that'instructors'can'play'in'a'student’s'
education"(Horn,"2007)."Imagine"that"you’re"the"instructor"quoted"in"Example"B"or"C"
in"Box"2."You’ve"noticed"a"“mismatch”"between"the"curriculum"that"you’re"supposed"
to"teach"and"the"students"in"your"classroom."You’ve"located"the"source"of"that"
mismatch"in"the"students’"abilities"(Example"B)"or"the"student’s"families"(Example"
C)."As"a"instructor,"what"can"you"do?"You"can’t"control"the"students’"abilities!"You"
can’t"control"their"families!"The"deficit"perspective"has"effectively"limited"your"
options."The"only"way"to"remedy"the"mismatch"is"to"“fix”"these"students,"say"by"
assigning"them"to"remedial"courses"as"discussed"above."Now,"to"be"clear,"students"
arrive"in"our"classes"with"very"different"mathematical"histories."However,"this"
heterogeneity"does"not"have"to"be"understood"from"a"deficit"perspective."
Suggestions"for"alternatives"are"discussed"in"Section"4"(see"also,"Adiredja,"2019b,"
2020;"Adiredja"et"al.,"2020;"Adiredja"&"Zandieh,"2020).""
The'deficit'perspective'works'with'culturefree'beliefs'of'STEM'to'prevent'
powerful'forms'of'cognition'from'entering'the'classroom'(Adiredja"&"Louie,"2020;"
Bang,"Warren,"Rosebery,"&"Medin,"2012;"Frade,"AciolyRégnier,"&"Jun,"2013).'There"
are"large"literatures"that"demonstrate"the"ways"that"both"cognition"(Cole,"2010;"
Hutchins,"1995a;"LCHC,"1983;"Radford,"2008)"and"mathematics"(Ascher,"2002;"
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Joseph,"2011;"Mukhopadhyay"&"Roth,"2012;"Peck,"2018;"Pinxten,"2016;"Pinxten,"Van"
Dooren,"&"Harvey,"1983)"are"not"culturefree,"but"rather"are"intertwined"with"
culture."However,"STEM"disciplines—especially"mathematics—are"often"seen"as"
universal"or"culturefree"(Bang"et"al.,"2012;"Ernest,"1998)."For"example,"just"before"
the"Example"D"in"Box"2,"the"quoted"policy"document"states:"“Among"the"many"
subjects"taught"in"school,"mathematics"is"probably"the"most"universal,"depending"
least"on"a"student's"background"and"culture”"(National"Research"Council,"1989,"p."
20)."Although"widespread,"these"beliefs"in"a"“culturefree”"mathematics"are"belied"
by"studies"of"mathematics"across"history"and"societies"(Peck,"2018),"which"reveal"
“various"types"of"mathematics,"irreducible"to"each"other”(Radford,"2008,"p."457)."
For"example,"Pinxten"and"colleagues"(Pinxten"et"al.,"1983)"describe"how"a"Navajo"
community"understood"space"very"differently"from"that"of"a"Western,"Euclidean"
perspective.""
Such"“culturefree”"beliefs"about"mathematics"can"be"damaging"for"students."
When"these"beliefs"about"mathematics"are"combined"with"deficit"perspectives"of"
students,"students’"outofschool"ways"of"knowing"and"doing"mathematics"are"
ignored"or"trivialized"(as"exemplified"by"Example"D"in"Box"2)."This"prevents"
powerful"forms"of"cognition"(e.g.,"the"Navajo"community’s"ways"of"understanding"
space"in"Pinxten"et"al.,"1983)"from"entering"the"classroom,"and"puts"students"in"
“untenable"epistemological"positions"that"work"against"engagement"in"meaningful"
learning”"(Bang"et"al.,"2012,"p."304;"see"also,"Bishop,"1990;"Frade"et"al.,"2013;"Shields"
et"al.,"2005).""
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Deficit'perspectives'work'to'keep'the'above'hidden,"preventing'critical'
introspection'on'the'part'of'educators'and'perpetuating'the'privilege'of'dominant'
groups"(Castro,"2012;"Frade"et"al.,"2013;"García"&"Guerra,"2004;"Horn,"2007;"Martin,"
2006,"2009;"Parsons,"2008;"Shields"et"al.,"2005)."Perhaps"the"most"pernicious"aspect"
of"deficit"perspectives"is"that,"by"locating"deficiencies"outside"of"the"school,"deficit"
perspectives"prevent"educators"from"seeing"their"role"in"the"systems"that"
perpetuate"oppression
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."Deficit"perspectives"thus"become"a"selffulfilling"prophesy,"
both"the"beginning"and"the"ending"of"a"descriptionexplanationprediction
prescription"cycle"(Valencia,"2010)"in"which:""
"
[F]irst"educators"describe"deficits,"deficiencies,"limitations,"and"shortcomings"
in"[students]"of"color"and"[students]"from"lowincome"homes;"next,"educators"
explain"these"deficits"by"locating"them"in"such"factors"as"limited"intelligence"
or"dysfunctional"families;"then,"educators"predict"the"perpetuation"and"
accumulation"of"the"deficits;"and,"finally,"educators"prescribe"educational"
interventions"designed"to"remediate"the"deficits"(Skrla"&"Scheurich,"2001,"p."
236,"p."236)"
"
As"we’ve"seen,"the"result"of"these"prescriptions"are"racialized"and"gendered"
experiences"in"mathematics"(Larnell,"2016;"Leyva,"Quea,"Weber,"Battey,"&"López,"
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"This"phrasing"is"important,"lest"we"fall"into"the"trap"of"applying"deficit"perspectives"to"educators."It"
is"not"my"intention"to"locate"fault"inside"educators"–"myself"included!"Rather,"the"fault"lies"in"the"
deficit"perspective"itself,"which"works"on"and"through"educators"to"structure"inequitable"educational"
opportunities."
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2019)"and"inequitable"access"to"educational"opportunities."Thus"deficit"perspectives"
lead"educators"to"differentially"structure"educational"opportunity"while"
simultaneously"obscuring"the"educator’s"role."This"in"turn"perpetuates"a"system"of"
privilege"and"oppression."
"It"could"be"otherwise."In"the"next"section,"I"describe"the"possibilities"that"can"
accrue"to"students,"instructors,"schools,"and"society"when"deficit"models"are"
abandoned.""
4. ANTIDEFICIT!EDUCATION!
Alternatives"to"deficit"perspectives"are"often"called"“antideficit”"or"“asset”"models"
(Adiredja,"2019b,"2019a;"Adiredja"et"al.,"2020;"Hagman,"2019)."In"this"paper,"I"use"
the"term"antideficit'to"draw"attention"to"the"active"work"that"it"takes"to"resists"
deficit"perspectives."Box"3"provides"a"definition"of"an"antideficit"perspective."
"
Definition:
Antideficit perspectives locate the source of academic problems within institutional
structures that work to limit access to educational opportunities. The focus is on the assets that
students bring to the classroom, rather than what they lack."
Box"3."Definition"of"the"antideficit,"assetbased"perspective"
"
Education"from"an"antideficit"perspective"repositions"student’s"histories,"
families,"communities,"and"cultures"as"sources"of"assets"rather"than"deficits'
(Adiredja,"2019a;"K."D."Gutiérrez,"Hunter,"&"Arzubiaga,"2009;"Moll,"Amanti,"Neff,"&"
Gonzalez,"1992;"Rosebery,"Ogonowski,"DiSchino,"&"Warren,"2010;"Turner,"Gutiérrez,"
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SimicMuller,"&"DíezPalomar,"2009)."What"this"means"is"that"students’"backgrounds"
are"brought"into,"rather"than"excluded"from,"the"classroom."The"resulting"
heterogeneity"is"leveraged"as"a"resource"for"all"students,"leading"to"opportunities"for"
understandings"that"include,"but"go"beyond,"those"prescribed"by"course"curricula.""
In"these"classrooms,"students"play"a"more"active"and"agential"role"in"their"
education."In"particular,"education"from"an"antideficit"perspective"positions"
students"as"epistemic"(knowledgebuilding)"agents"and"ontological"(worldbuilding)"
agents."Epistemic"agency"involves"giving"students"the"authority"to"make"and"share"
meaning"in"the"classroom"(Bang"et"al.,"2012;"Burton,"1995;"Owens,"2015;"Valencia,"
2010)."Within"the"undergraduate"mathematics"community,"this"has"been"
popularized"through"InquiryBased'Mathematics'Education'(IBME)'(Ernst,"Hodge,"&"
Yoshinobu,"2017;"Laursen"&"Rasmussen,"2019)."IBME"(and"other"forms"of"active"
learning"(Freeman"et"al.,"2014)"may"be"particularly"powerful"for"members"of"
traditionally"oppressed"groups"(Laursen,"Hassi,"Kogan,"&"Weston,"2014;"Theobald"et"
al.,"2020),"and"can"be"a"powerful"tool"for"antideficit"teaching"(Tang"et"al.,"2017)"(but"
that"potential"is"not"automatically"realized,"Johnson,"Keller,"AndrewsLarson,"
Fortune,"&"Keene,"2020)."Ontological"agency"involves"positioning"students"as"actors"
who"can"use"mathematics"to"both"understand"the"world"from"a"critical"perspective"
and"author"new"roles"for"themselves"in"new,"more"socially"just"visions"the"world
4
"
(K."D."Gutiérrez"et"al.,"2009;"Gutstein,"2006;"Turner"et"al.,"2009)."Within"the"
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"It"should"be"clear"that"students"don’t"need"instructors’"“permission”"to"act"agentially"in"the"world."
Students"are,"and"always"have"been,"epistemic"and"onotological"actors."It’s"just"that"institutions"often"
work"to"suppress"these"forms"of"agency"in"the"classroom,"especially"for"students"in"oppressed"
groups."What"I’m"arguing"for"here"are"classroom"practices"that"foreground"and"support"student"
agency,"but"I"don’t"mean"to"suggest"that"students"cannot"act"agentially"in"the"absence"of"such"
practices."""
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mathematics"education"community,"this"has"been"taken"up"largely"though"
Mathematics'for'Social'Justice"(Gutstein"&"Peterson,"2006;"Karaali"&"Khadjavi,"2019;"
Wager"&"Stinson,"2012)"(see"also,"the"Special"Issue"of"PRIMUS"on"Mathematics"for"
Social"Justice,"Volume"29"Issue"34)".""
There"are"two"tensions"here."With"respect"to"epistemic"agency"there"is"a"
tension"between"giving"students"epistemic"agency"and"ensuring"that"students"learn"
institutionallyvalued"disciplinary"knowledge."With"respect"to"ontological"agency,"
there"is"a"tension"between"allocating"limited"classroom"time"to"teaching"the"STEM"
curriculum"and"to"engaging"students"in"worldbuilding,"social"justice"activity"
(Gutiérrez"(2009)"calls"this"“playing"the"game”"and"“changing"the"game”)."Different"
authors"negotiate"these"tensions"in"various"ways."It’s"beyond"the"scope"of"this"
review"to"discuss"these"in"depth,"but"suffice"it"to"say"that"the"tensions"are"essential"
to"teaching"from"an"antideficit"perspective"(R."Gutiérrez,"2006,"2007)."For"a"reviews"
of"the"tension"between"epistemic"agency"and"disciplinary"knowledge,"see"(Nasir,"
Hand,"&"Taylor,"2008)"and"(Radford,"2012)."For"discussions"of"the"tension"between"
teaching"STEM"and"engaging"students"in"worldbuilding,"social"justice"activity"see"
(Gutstein,"2006)"and"(R."Gutiérrez,"2007).""
Regardless"of"how"these"tensions"are"negotiated,"it"should"be"clear"that"
education"from"an"antideficit"perspective"involves"changes"to"classroom"practices"
and"changes"to"how"mathematics"itself"is"conceptualized."Classrooms"become"active"
and"discursive"spaces"for"making"meaning,"and"disciplinary"knowledge"becomes"
terrain"to"be"analyzed"and"contested"(Frade"et"al.,"2013;"Owens,"2015;"Peck,"2018)."
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It"takes"work"to"make"these"changes."First,"instructors"must"(a)"examine"how"
institutional"practices"operate"in"ways"that"perpetuate"privilege"and"oppression,"
and"how"deficit"perceptions"obscure"these"practices"(García"&"Guerra,"2004);"(b)"
become"aware"of"their"own"agency"to"change"these"practices"(Shields"et"al.,"2005);"
(c)"understand"that"all"ways"of"knowing—including"those"that"are"institutionally"
valued—are"cultural"(Bryan"&"Atwater,"2002)"and"that"disciplinary"knowledge"is"
not"“settled”"but"rather"is"constantly"being"contested"by"practitioners"(Bang"et"al.,"
2012;"Ernest,"1994,"1998);"and"(d)"learn"to"teach"in"ways"that"are"“relevant"and"
responsive”"to"students’"social"realities,"their"cultural"practices,"and"their"identities"
(Martin,"2007,"p."18;"LadsonBillings,"1991,"1999;"A."J."Rodriguez"&"Kitchen,"2005),"
including"ways"of"incorporating"heterogeneity"as"a"resource"(e.g.,"Adiredja"et"al.,"
2020).""
In"addition,"instructors"must"build"real,"human"relationships"with"students,"
including"understanding"students’"unique"histories."This"is"important"for"many"
reasons."First,"it"is"essential"if"instructors"are"going"to"leverage"diversity"without"
reducing"students"to"stereotypical"traits"based"on"their"membership"in"social"
categories"like"gender,"race,"and"ethnicity"(K."D."Gutiérrez"&"Rogoff,"2003)."Second,"it"
allows"instructors"to"structure"activities"that"are"not"only"relevant"to"students’"
histories,"but"also"to"their"futures"(Shields"et"al.,"2005)."Finally,"it"allows"for"more"
expansive"versions"of"learning,"one"which"rejects"individuation"as"the"goal"of"
education"and"instead"conceptualizes"education"as"a"social"project,"where"we"“come"
together"to"recognize"ourselves"as"historical"and"political"beings"and"where"we"
15"
critically"labour"together"to"make"the"common"world"a"better"place"for"all”"(Radford,"
2012,"p."116;"see"also"Noddings,"1992).""
Below,"I"will"share"some"examples"of"how"I"try"to"engage"in"this"work"in"my"
own"practice."Although"I"present"these"ideas"at"a"high"level"of"generality,"I"do"not"
mean"to"suggest"that"all"ideas"work"the"same"for"all"students."Indeed,"I"would"
suggest"the"opposite."Because"students"have"diverse"histories"with"mathematics,"it"
is"important"that"instructors"are"attentive"to"these"histories"in"all"classroom"
interactions."This"is"likely"of"particular"importance"for"students"from"systemically"
oppressed"groups,"who"must"be"understood"as"individuals"but"whose"membership"
in"oppressed"social"categories"often"influences"their"mathematical"experiences"
(Larnell,"2016;"Leyva"et"al.,"2019)."
4.1. Example!1:!Dismantling!rigid!course!sequences.!
In"university"math"departments,"courses"are"often"structured"in"a"rigid"
hierarchical"sequence,"in"which"access"to"one"course"is"dependent"on"passing"a"“pre
requisite”"course."Incoming"students"are"placed"into"this"sequence"based"on"a"
placement"test."In"some"cases,"students"must"take"multiple"semesters"of"remedial"
(noncreditbearing)"courses"just"to"access"their"first"creditbearing"math"class"
(Bailey,"Jeong,"&"Cho,"2010)."When"students"are"placed"into"remedial"courses,"they"
are"more"likely"to"dropout"before"earning"their"degree"(Ganga,"Mazzariello,"&"
Edgecombe,"2018).""
Placement"into"this"sequence"is"not"independent"of"privilege"(Larnell,"2016)."
Students"from"oppressed"groups"are"overrepresented"in"remedial,"non"credit
16"
bearing"courses,"where"they"accumulate"debt"but"often"do"not"get"a"degree"(Bailey"
et"al.,"2010)."Meanwhile"students"from"privileged"backgrounds"are"more"likely"to"be"
placed"into"creditbearing"courses,"and"go"on"to"graduate."In"this"way,"this"system"of"
rigid"course"sequencing"works"to"perpetuate"privilege"and"oppression."At"the"same"
time,"the"workings"of"the"system"are"obscured"behind"deficit"perspectives."Because"
placement"is"done"through"supposedly"neutral"means"(most"often,"a"standardized"
exam—which,"as"I"address"below,"artificially"constrain"the"kinds"of"knowledge"that"
students"can"demonstrate),"the"fault"for"placement"in"remedial"classes"can"be"
located"inside"of"students"(e.g.,"“students…"are"not"prepared,”"from"Example"A"in"
Box"2).""
However,"when"students"are"given"the"right"kinds"of"support,"many"students"
who"would"otherwise"be"assigned"to"remedial"courses"can"be"successful"in"credit
bearing"courses—they"can"“jump"the"sequence.”"One"powerful"form"of"support"is"co
requisite"courses."In"this"system,"the"rigid"system"of"prerequisites"is"dismantled."
Students"are"placed"into"creditbearing"courses"immediately"upon"enrollment."At"
the"same"time,"they"are"enrolled"in"supportive"corequisite"courses"(Complete"
College"America,"2016).""
In"my"department"at"the"University"of"Montana,"we"have"undertaken"a"
corequisite"initiative'to"place"students"in"creditbearing"courses"immediately,"while"
providing"supportive"corequisite"courses."So"far,"we"have"implemented"this"system"
for"College"Algebra"and"Precalculus,"and"we"are"piloting"corequisite"courses"for"
Calculus"I."This"work,"spearheaded"by"my"colleague"Laruen"Fern,"has"been"guided"by"
the"Math'Pathways'initiative"from"the"Charles"M."Dana"Center"at"the"University"of"
17"
Texas"(see"(Complete"College"America,"2016)"and"(“Charles"A."Dana"Center:"Co
requisite"support"materials,”"n.d.)"for"resources)."Corequisite"courses"meet"once"or"
twice"per"week."They"do"not"have"additional"homework"and"are"graded"as"
credit/nocredit."The"courses"are"taught"by"the"same"instructor"as"the"credit
bearing"course."The"content"of"the"corequisite"class"is"designed"to"be"responsive"to"
the"creditbearing"course,"such"that"the"corequisite"courses"helps"students"more
fully"participate"in"their"creditbearing"class"that'week."This"is"called"justintime"
preparation."In"this"way,"the"corequisite"courses"are"tightly"coupled"with"their"
related"creditbearing"courses."Our"results"have"been"encouraging,"with"a"
substantial"reduction"in"remedial"course"taking,"and"an"increase"in"student"success"
in"creditbearing"courses."Departments"across"the"country"have"seen"similar"results"
(Complete"College"America,"2016).""
4.2. Example!2:!More!expansive!forms!of!summative!
assessment!
Educators"differentiate"between"formative'uses"of"assessment"and"summative'
uses."When"assessments"are"used"formatively,"they"are"used"to"guide"teaching"and"
learning."When"assessments"are"used"summatively,"they"are"used"to"evaluate"
learning."In"this"example,"I"will"focus"on"summative"assessment."In"undergraduate"
mathematics"classrooms,"summative"assessments"often"have"a"singular"form:"
pencilandpaper"testing.""Traditional"pencil"and"paper"tests"are"often"treated"as"the"
de'facto"means"by"which"students"display"their"knowledge"for"evaluation"purposes."
18"
In"many"departments,"testing"is"so"ubiquitous"that"it"doesn’t"even"feel"like"a"
decision."
There"are"reasons"to"question"the"unquestioned"ubiquity"of"testing."In"many"
ways,"traditional"pencilandpaper"tests"narrowly"constrain"the"sorts"of"knowledge"
that"can"be"assessed"and"the"means"by"which"it"can"be"displayed."Such"tests"are"
individualistic"even"though"problem"solving"is"often"collaborative,"and"proof"is"
inherently"social"(Lakatos,"1979;"MacKenzie,"1999)."Tests"restrict"access"to"
resources,"even"though"mathematical"cognition"often"incorporates"tools,"resources,"
and"other"features"of"the"material"and"cultural"environment"(Bass,"2011;"Hutchins,"
1995b;"Lave,"1988;"Taylor,"2009;"Wertsch,"1998)."Tests"are"timeconstrained,"even"
though"mathematical"work"is"slow"and"thoughtful,"and"often"requires"periods"of"
downtime"(Hadamard,"1945;"Liljedahl,"2009)."Tests"require"responses"to"fixed"
problems,"even"though"mathematical"competence"is"multifaceted"and"there"are"
many"ways"to"show"mastery"of"a"concept."Traditionally,"students"have"only"one"
chance"to"take"a"test,"even"though"mathematical"work"involves"making"and"learning"
from"mistakes"(Liljedahl,"2009)."Altogether,"tests"privilege"a"very"narrow"slice"of"
mathematical"competence"while"ignoring"vast"amounts."The"erasure"of"expertise"
may"be"more"profound"for"students"from"oppressed"communities,"who"often"have"
rich"histories"of"communitybased"mathematical"practices"that"look"very"different"
from"pencilandpaper"tests"(Darling,"2016;"Taylor,"2009)."
An"antideficit"approach"incorporates"more"expansive"forms"of"summative"
assessment."In"my"own"practice,"I"use"studentgenerated"portfolios"as"the"primary"
means"by"which"students"demonstrate"their"mastery"of"the"course"objectives."I"
19"
provide"students"with"a"set"of"objectives"and"a"developmental"progression"that"
describes"increasing"levels"of"mastery"for"each"objective."Throughout"the"unit,"
students"selfassess"their"mastery"of"the"objective"according"to"the"developmental"
progression."At"the"end"of"the"unit,"students"choose"the"work"that"best"demonstrates"
their"mastery"and"write"an"argument"for"why"this"work"demonstrates"mastery"of"
the"objective."They"can"choose"work"produced"individually"or"collectively,"in"or"
outside"of"class,"with"or"without"external"resources."For"students"who"prefer"a"
traditional"test,"I"provide"a"set"of"questions"that"can"function"as"a"takehome"test"
(students"still"must"write"an"argument"for"how"their"work"on"the"questions"shows"
their"mastery"of"the"objectives)."After"they"get"feedback,"students"can"revise"and"
resubmit"their"portfolios"an"unlimited"number"of"times."For"examples,"see"Appendix"
A.""
Portfolios"are"powerful"tools"for"assessment"from"an"antideficit"perspective,"
because"they"broaden"the"forms"of"knowledge"that"can"be"assessed"and"the"means"
by"which"it"can"be"displayed."Whereas"tests"are"individualistic"and"restrict"access"to"
resources,"portfolios"allow"students"to"submit"work"that"was"produced"
collaboratively"and"with"resources."Whereas"tests"are"timeconstrained,"portfolios"
provide"an"opportunity"for"students"to"submit"slow,"thoughtful"work."Whereas"tests"
require"students"to"respond"to"fixed"problems,"portfolios"give"students"flexibility"in"
choosing"how"they"demonstrate"mastery"of"a"concept.""Whereas"tests"often"do"not"
allow"for"revision,"portfolios"provide"an"opportunity"for"students"to"learn"from"their"
mistakes"through"the"revision"process.""
20"
Portfolios"are"not"the"only"way"to"broaden"assessment"practices"for"anti
deficit"teaching."One"alternative"to"traditional"onetime"testing"that"is"gaining"
currency"in"the"field"is"“mastery"grading,”"(See"upcoming"special"issue"of"PRIMUS)."
Mastery"Grading"uses"individual"pencilandpaper"testing"for"summative"
assessment,"but"with"two"crucial"differences"from"traditional"testing:"(1)"students"
are"graded"based"on"their"demonstrated"mastery"of"an"objective"using"pre
determined"criteria,"rather"than"the"percentage"of"points"earned,"and"(2)"students"
are"allowed"to"reassess"to"demonstrate"improved"mastery."From"an"antideficit"
perspective,"mastery"grading"is"powerful"because"it"allows"students"to"make"and"
learn"from"mistakes,"and"because"it"can"alleviate"the"time"pressure"of"traditional"
testing."However,"because"mastery"grading"generally"relies"on"traditional"paper
andpencil"tests,"it"can"still"constrain"the"kinds"of"knowledge"that"can"be"assessed"
and"the"means"by"which"students"can"express"it."For"examples"of"more"expansive"
forms"of"summative"assessment,"see:"(Brilleslyper"et"al.,"2012;"Gold,"Keith,"&"
Marion,"1999;"Hanusch,"2019;"Huntley"&"Flores,"2011;"Omar,"Karakok,"Savic,"
Turkey,"&"Tang,"2018;"Rash,"1997)."
"
4.3. Example!3:!Landscapes!of!investigation!
In"our"graduate"program"for"math"teachers,"we"have"some"math"content"
courses"that"are"taken"by"elementary,"middle,"and"high"school"teachers,"all"in"the"
same"course."Some"participants"have"an"undergraduate"math"degree,"including"
experience"with"abstract"mathematics"and"formal"proof."Others"may"have"deep"
21"
knowledge"of"elementary"mathematics,"but"little"experience"with"the"traditional"
undergraduate"mathematics"curriculum."Thus,"these"classes"have"a"great"
heterogeneity"in"terms"of"students’"background."
Antideficit"teaching"requires"giving"all"students"an"opportunity"to"develop"as"
mathematicians."With"this"level"of"heterogeneity"of"background,"it"would"not"be"
appropriate"to"assign"narrow"mathematical"exercises,"as"these"would"not"be"open"
enough"to"accommodate"the"heterogeneity"in"the"class."Instead,"I"assign"landscapes'
of'investigation"(see"Skovsmose"(2001)"for"a"detailed"description"and"Appendix"B"
for"an"example)."Students"are"given"a"scenario"or"mathematical"object."From"there,"
they:"""
1. Pose'questions"that,"for"them,"require"problem"solving"(that"is,"the"
students"do"not"know"the"answer,"nor"do"they"know"of"a"method"to"find"
the"answer;"Bolognese"and"Steward"(2017)"call"these"“level"3”"questions.)"
2. Explore'those'questions"by,"e.g.,"noticing,"imagining,"playing,"conjecturing,"
exploring,"creating,"visualizing,"generalizing,"verifying,"explaining,"and"
justifying""
3. Produce'a'writeup,'written"with"their"colleagues"in"mind."The"writeup"
should"be"understandable"to"everyone"in"the"class.'
4. Provide'a'peer'review,"in"which"class"members"give"their"colleagues"
feedback"on"the"following"questions:"Do"I"find"the"work,"including"the"
questions"that"are"being"explored,"to"be"clear"and"understandable?;"Why"
is"the"reasoning"sensible"from"the"author’s"perspective?;"Would"the"
mathematical"community"be"likely"to"accept"the"reasoning?;"and"What"
22"
connections"do"I"see"in"the"work,"and"what"further"questions"can"be"
explored,"based"on"the"work?"(Radford,"2012)"
5. Revise'the'investigation'based"on"peer"feedback."""
Landscapes"of"investigation"are"powerful"tools"in"antideficit"teaching"for"at"
least"three"reasons."First,"they"both"accommodate"heterogeneity"and"leverage"it"as"a"
resource."In"steps"1"and"2,"landscapes"of"investigation"accommodate"heterogeneity."
Because"students"can"pose"their"own"questions"(step"1),"they"are"each"working"on"
challenging"mathematics"in"step"2,"even"though"that"can"look"very"different"for"
different"students."In"steps"3"and"4,"landscapes"of"investigation"leverage"
heterogeneity"as"a"resource,"as"students"have"to"write"for,"and"get"feedback"from,"a"
diverse"audience."In"doing"so,"landscapes"of"investigation"position"students"as"
epistemic"agents"with"the"ability"to"make,"share,"and"contest"mathematics.""
Second,"landscapes"of"investigation"provide"students"with"multiple"ways"to"
show"mathematical"competence,"including:"asking"good"questions,"being"creative,"
visualizing"complex"ideas,"developing"and"using"representations"and"notations,""
writing"cogent"and"convincing"explanations,"reading"graciously"and"skeptically,"and"
making"connections"(c.f.,"Horn,"2012).""
Third,"landscapes"of"investigation"help"to"foster"more"expansive"forms"of""
learning,"including"developing"empathy,"courage,"and"humility."Producing"and"
reading"a"writeup"involves"perspective"taking,"which"fosters"empathy."When"
producing"a"writeup,"students"must"take"the"perspective"of"a"reader,"and"consider"
how"their"text"will"be"understood"by"people"with"different"backgrounds"from"their"
own."When"reading"a"writeup,"peer"reviewers"must"take"the"perspective"of"the"
23"
authors"(“why"is"the"reasoning"sensible"from"the"author’s"perspective?”),"and"of"the"
mathematical"community"(“would"the"mathematical"community"be"likely"to"accept"
the"reasoning?”)."Sharing"the"writeups"with"others"is"risky,"and"thus"doing"so"
develops"courage"and"humility"(Lampert,"1990;"Radford,"2012).""
4.4. Example!4:!Building!human!relationships!
Antideficit"teaching"requires"authentic"relationships"between"teachers"and"
students."In"the"beginning"of"the"semester,"I"use"three"techniques"to"get"to"know"my"
students"(there"are"many"other"techniques,"these"are"examples"that"have"worked"for"
me).""
1. I"ask"students"to"write"an"autobiography"during"the"first"week"(Braun,"2014;"
Pinter,"2014)."I"ask"them"to"address"their"history"and"relationship"with"
mathematics,"to"share"something"that"they"are"proud"of,"and"to"describe"their"
desired"future"trajectory.""
2. I"invite"students"to"have"a"lowstakes,"written"conversation"with"me"via"a"
“conversation"form.”"(based"on"work"by"Van"Der"Werf,"in"Horn,"2017,"pp."36–
37)."The"form"has"two"spaces"for"each"day"of"the"first"week."The"first"space"is"
for"the"student"to"write"something,"and"the"second"space"is"for"my"response."
On"the"first"day"of"class"I"give"students"a"sheet"of"paper"that"is"blank"on"one"
side"and"which"has"the"conversation"form"printed"on"the"other."I"ask"students"
to"fold"the"paper"into"a"name"tent,"with"the"form"on"the"interior."This"is"their"
name"tent"for"the"first"few"days"of"class."In"the"last"five"minutes"of"class,"I"ask"
24"
students"to"write"me"a"note"in"the"slot"for"“day"1.”"I"then"respond"to"each"
note,"and"we"continue"like"this"for"the"rest"of"the"week.""
3. It"is"difficult"for"me"to"learn"students’"names."I"openly"admit"this"to"students."
At"the"same"time,"I"strive"to"learn"every"name"within"two"class"sessions."To"
do"that,"I"take"a"class"photo"with"everyone"holding"their"name"tents"on"the"
first"day,"and"I"study"this"photo"for"the"rest"of"the"week."When"I"am"
responding"to"a"student"on"the"conversation"form,"I"find"them"in"the"class"
picture."When"I"read"their"autobiographies,"I"find"their"conversation"form"
and"their"picture."In"this"way"I"am"able"to"associate"names,"faces,"histories,"
and"futures.""
"
When"I"know"a"student’s"history"with"mathematics,"I"can"structure"the"class"
to"be"mindful"of"that"history."For"example,"many"undergraduate"preservice"
elementary"teachers"report"negative"histories"with"school"mathematics."For"these"
students,"math"has"often"meant"“getting"correct"answers"following"rote"procedures"
under"time"pressure.”"Thus,"an"emergent"goal"of"my"class"with"them"is"to"foster"a"
more"positive"conception"of"mathematics"as"a"creative"and"sensible"human"activity.""
When"students"share"what"they"are"proud"of,"I"can"try"to"incorporate"some"of"
these"strengths"into"the"class."For"example,"one"year"I"had"many"students"describe"
their"artistic"accomplishments."We"therefore"did"a"project"in"which"we"explored"
patterns"in"geometric"artwork"and"created"our"own"pieces.""
When"I"know"a"student’s"future,"I"can"work"with"them"to"align"the"course"
assignments"and"assessments"to"that"future."For"example,"we"have"a"course"for"
25"
graduate"students"in"mathematics"called"Teaching"College"Math."Students"in"the"
course"have"a"variety"of"imagined"futures,"including"teaching"at"2year"institutions,"
4year"teachingintensive"institutions,"and"researchintensive"institutions."A"student"
aiming"for"a"2year"institution"may"be"interested"in"resources"for"teaching"calculus"
(e.g.,"Driskell"&"Malagon,"2013;"Oehrtman,"2008;"Speiser"&"Walter,"2008;"Thompson"
&"Silverman,"2008),"a"graduate"student"in"Algebra"may"be"interested"in"learning"
more"about"resources"for"Abstract"Algebra"(e.g.,"Cook,"2015;"Larsen,"2013;"Larsen"&"
Lockwood,"2013),"and"a"student"in"statistics"might"be"more"interested"in"resources"
for"undergraduate"statistics"(e.g.,"GAISE"College"Report"ASA"Revision"Committee,"
2016;"Garfield,"DelMas,"&"Zieffler,"2012;"Hathaway,"1994)."Thus,"in"addition"to"
learning"about"general"principles"for"evidencebased"teaching"(Schoenfeld,"2014),"I"
also"provide"resources"specific"to"a"variety"of"courses,"and"assignments"that"enable"
students"to"engage"with"the"resources"of"their"choosing."
Building"relationships"with"students"helps"me"align"courses"to"their"histories,"
their"strengths,"and"their"futures."It"also"helps"to"foster"belongingness—the"sense"
that"one"is"cared"for"by"members"of"the"community"(Horn,"2017)."While"the"
strategies"listed"above"help"to"foster"belongingness"in"the"beginning"of"the"year,"the"
most"important"strategy"is"to"simply"be"present,"listen"to"students,"and"affirm"their"
experiences."If"a"student"needs"to"tell"me"a"story"about"why"they"were"absent,"why"
they"need"an"extension,"or"why"they"feel"joyful"or"frustrated,"I"always"make"time"for"
it."I"have"taken"long"walks"around"campus"with"students,"as"they"tell"me"stories—of"
hardship,"of"friendship,"of"kinship."Whatever"is"the"reason"that"they"want"to"talk,"I"
make"sure"that"I"prioritize"it."Students"recognize"these"caring"interactions."More"
26"
than"once,"students"have"told"me"that"they"likely"would"have"droppedout"had"I"not"
been"so"receptive"to"them."
Caring"for"students"can"be"difficult"in"the"rush"of"a"semester."It"can"be"
tempting"to"reduce"students"to"only"their"mathematical"behaviors—often,"as"
discussed"above,"only"the"very"narrow"set"of"those"behaviors"that"become"inscribed"
on"tests."As"Noddings"(1988)"explains,"“In"every"human"encounter,"there"arises"the"
possibility"of"a"caring"occasion."(…")"Teaching"is"filled"with"caring"occasions,"or,"
quite"often,"with"attempts"to"avoid"such"occasions”"(p."222)."Antideficit"teaching"
means"insisting"on"treating"every"occasion"as"a"caring"occasion,"and"on"treating"
every"student"as"a"full"human
5
.""
For"more"examples"of"antideficit"work"in"real"undergraduate"mathematics"
courses,"see:"(Adiredja,"2019b,"2020;"Adiredja"et"al.,"2020;"Adiredja"&"Zandieh,"
2020;"K."D."Gutiérrez"et"al.,"2009;"Moreno"&"Rutledge,"2017)."
5. CONCLUSION!
I"began"this"review"by"focusing"on"achievement."I"showed"how"deficit"perspectives"
work"to"structure"differential"patterns"of"achievement,"conscripting"students"and"
instructors"to"narrow"roles"that"perpetuate"inequality."I"then"presented"an"alternate"
vision"of"education"from"an"antideficit"perspective."This"perspective"honors"
"""""""""""""""""""""""""""""""""""""""""""""""""""""""
"
5
"It"should"go"without"saying"that,"even"though"I"have"discussed"the"pedagogical"benefits"of"getting"to"
know"students,"the"primary"purpose"is"not"transactional"but"rather"relational."Moreover,"it"is"
important"to"note"that"institutional"structures"can"produce"power"imbalances"between"instructors"
and"students,"and"these"can"interact"with"other"societallyproduced"imbalances,"such"as"if"instructors"
and"students"are"of"different"genders."Instructors"should"be"mindful"of"these"imbalances,"without"
allowing"a"fear"of"imbalance"to"prevent"caring"interactions.""
27"
students"and"their"histories"and"leverages"diversity"to"create"new"opportunities"and"
more"expansive"forms"of"learning.""
I"hope"that"this"piece"serves"as"an"invitation"to"those"who"are"new"to"the"
work"of"antideficit"teaching,"and"as"encouragement"to"those"who"are"engaged"in"it."
Given"the"difficulty"of"the"work,"we"must"sustain"each"other."In"that"spirit,"I"would"
like"to"conclude"by"sharing"two"quotations"that"have"recently"nourished"my"
commitment"to"the"work."The"first"comes"from"the"philosopher"Simone"Weil;"I"was"
made"aware"of"the"quote"by"Francis"Su"(Su"&"Jackson,"2020)."The"second"comes"
from"the"educator"Maxine"Green;"I"was"made"aware"of"the"quote"by"Ilana"Horn"
(Horn,"2019)."
"
“Every"being"cries"out"silently"to"be"read"differently”"(Simone"Weil)"
"
In"this"powerful"statement,"Simone"Weil"reminds"us"that"people"are"often"
judged"in"ways"that"diminish"them."All"of"us,"and"all"of"our"students,"have"powerful"
histories"with"and"diverse"repertoires"for"mathematics."We"all"deserve"to"be"seen"
for"these"strengths,"but"too"often"we"reduce"our"students"to"their"deficits."Let"us"
hear"our"students’"silent"cries"to"be"read"differently."Let"us"read"our"students"from"
an"antideficit"perspective.""
This"is"hard"work,"because"all"too"often"the"structures"and"normative"
practices"of"our"institutions"seem"to"facilitate"deficit"perspectives"of"students."It"is"
difficult"to"change"institutional"structures,"difficult"to"work"against"normative"
practice."The"good"news"is"that"these"structures"and"practices"are"not"set"in"stone."
28"
As"I"write"this,"the"normative"practices"of"many"institutions"have"been"massively"
disrupted"by"a"global"pandemic."This"rupture"in"the"takenforgranted"exposes"how"
institutions"are"produced"by"us,"and"can"be"changed"by"us."Rather"than"be"
constrained"by"“what"is,”"then,"let"us"imagine,"
""
“what"could"be?”"(Maxine"Green)"
"
What"could"be"if"we"read"students"differently?""
What"could"be"if"we"expanded"our"notions"of"mathematical"competence,"so"
that"we"could"see"it"in"all"of"our"students?""
What"could"be"if"we"reorganized"our"institutions"to"focus"on"students’"
strengths,"rather"than"their"deficits?""
What"could"be?""
I"hope"you"will"join"me"in"imagining"what"could"be,"and"in"working"towards"a"
more"just,"humane,"and"expansive"future.""
!"
29"
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" "
42"
7. APPENDIX!A:!EXAMPLE!PORTFOLIOS!
Course: Number and operations for K8 teachers
Objective: I understand the structure of fractions from a units
perspective
Not there yet
Approaching mastery
Mastery
I mostly or solely conceptualize
fractions from a “pieces out of
pieces” perspective:
• I mostly conceptualize th e
numerator and denominator as
“numbers of pieces,”
• I mostly conceptualize a
fraction as a specified number
of pieces out of a total number
of pieces (e.g., !
" as “three
pieces out of four pieces”)
I can coordinate two levels of units:
• I can create u nit fraction s given
a whole
• I can create a whole given a
unit fraction
• I can show how fractions are
composed of unit fractions
(i.e., “expanded form”)
• I can explain the limitations of
an “out of” perspective.
I can coordinate three levels of
units (All of “approaching,” plus):
• I can show multiple ways to
create fractions, some of which
do not require partitioning the
entire whole (e.g., I can create
3/4 withou t partitioning a
whole into four pieces)
• I can create any fraction from
any other fraction (e.g., given a
bar that represents 3/5, I c an
find the bar that represents
4/3).
Find examples: Find examples in your work that demonstrate that you understand the
structure of fractions from a units perspective, or create new pieces from the
supplemental questions
Make an argument: Explain how the piece(s) you chose demonstrates your mastery of
this objective:
43"
Course: Calculus I
Objective: I understand accumulator functions.
Not there yet
Approaching mastery
Mastery
I do not understand integrals with
variable endpoints, or how they
relate to antiderivatives
• I can explain the meaning of
functions written in the form,
#$%&' ( )$*&+*
,

• I can evaluate accumulator
functions:
o Graphically, using geometry
o Numerically, using a calculator
o Analytically, using the FTC
• I can differentiate accu mulator
functions of the form,
#$%&' ( )$*&+*
,

All of “approaching,” plus:
• I can find the antiderivative of
any function
• I can differentiate accu mulator
functions of the form,
#$%&' ( )$*&+*
.$,&

Find examples: Find examples in your work that demonstrate that you understand
accumulator functions, or create new pieces from the supplemental questions
Make an argument: Explain how the piece(s) you chose demonstrates your mastery of
this objective:
"
" "
44"
8. APPENDIX!B:!EXAMPLE!LANDSCAPE!OF!
INVESTIGATION!
Note:"The"example"below"comes"from"a"course"on"algebra"for"teachers."In"the"
course,"we"defined"algebra"as"follows:"“Algebra"is"the"human"activity"of"
algebratizing:"structuring"the"world"(including"the"mathematical"world)"by"finding"
relationships,"generalizing,"abstracting,"and"justifying.”"
"
"
Invitation:!Staircase!numbers!a""
"
"
Investigation:"Pose"questions"about"staircase"
numbers"and"algebratize.'
"
a"From"Josh"Zucker"at"the"Julia"Robinson"Math"Festival"
"
5
Staircase Numbers
suggested by Joshua Zucker, Julia Robinson Mathematics Festival
EXTENSION:
Which numbers can’t be expressed as the sum of two or more consecutive positive
integers?
Which numbers can’t be expressed as the sum of three or more consecutive positive
integers?
Find more Julia Robinson Mathematics Festival problem sets at jrmf.org/problems.php.