1. formative assessment in mathematics
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7/27/2019 1. Formative Assessment in Mathematics
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Formative Assessment in MathematicsAuthor(s): Margaret E. McIntoshSource: The Clearing House, Vol. 71, No. 2, Forms and Functions of Formative Assessment(Nov. - Dec., 1997), pp. 92-96Published by: Taylor & Francis, Ltd.
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FormativeAssessmentnMathematics
MARGARETE. McINTOSH
I
Sam bigfanof usingdefinitions s a starting oint or
thinkingabout a topic, so let us look at a definitionof
assessment from the National Council of Teachersof Math-
ematics(1995): "Assessment s ... theprocessof gatheringevidence about a student'sknowledgeof, abilityto use, and
dispositiontoward,mathematicsand of makinginferences
fromthatevidence for a varietyof purposes" 3).
Dependingon your age, this definitionmay describethe
experience you had with assessmentin mathematicsduring
yourschool career,but for most readers,"testing"was real-
ly the only kind of "assessment"we knew.Like clockwork,at the end of every few sections of our math book, there
would be a quiz (for a GRADE) and at the end of every
chapter, here would be a TEST (for a MAJORGRADE).Then, no matterwhat gradesany of us received,we would
go off to the next chapter,wherethe cycle beganagain.Thattypeof testing(of which there aremanyvarieties) s
known in today's parlance as "summativeassessment,"defined as "a culminatingassessment, which gives infor-
mation on student'smasteryof content"(Association for
SupervisionandCurriculumDevelopment1996, 60). Table
1 summarizes the principal characteristicsof summative
assessment.
In contrast, he focus of this special issue of The Clear-
ing House is "formativeassessment" also see table 1). For
thepurposes
of thisarticle,
thefollowing
definition of for-
mative assessment will be used: "assessment which pro-vides feedback to the teacher for the purposeof improvinginstruction"ASCD 1996, 59). This conceptof assessment
meshes nicely with the NCTM definitionshown above.
Formativeassessment-with or without that name-has
always been found in the classroom, to an extent that
dependedon individual teachers'attitudes toward assess-
MargaretE. McIntosh s a professorin theDepartmentofCurriculum ndInstruction,College of Education,Univer-
sity of Nevada,Reno and a consultingeditor or TheClear-
ing House.
ment. The teacher who believes, as GrantWiggins does,that "good teaching is inseparablefrom good assessing,"
already uses an ongoing cycle of teaching, assessment,assessment of the teaching, reteaching (as necessary),assessment,teaching,andso on. "Assessment should serve
as the essentiallinkamongcurriculum,eaching,and learn-
ing"(Wilcox andZielinkski1997, 223).
Although there can be overlapbetween some types of
formativeand summativeassessments,and althoughthere
are both informaland formalmeans to assess students,in
this article,I will primarilyoffer suggestions for informal
formativeassessment for the mathematicsclassroom. The
ideaspresentedwill fall into threecategories (adapted rom
Clarkeand Wilson 1996):
TABLE1Characteristics of Summative and Formative
Assessment
Summative Formativeassessment assesssment
Time At the conclusion f Duringa learninga learningctivity activity
Goal To makea decision Toimprove
earn-
ing
Feedback Finaludgment Returnomaterial
Frame f Sometimes orma- Alwayscriterionreference tive(comparingach (evaluatingll
student gainst ll students ccord-theothers);ome- ingto the sametimescriterioneval- criteria)uating achstudent
accordingo thesamecriteria)
Source:AdaptedromR. Pregent. 994.ChartingYourCourse:How to Prepareto TeachMoreEfficiently.Madison,Wis.:MagnaPublications,nc.
92
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Vol.71,No.2 FormativeAssessment inMath 93
FIGUREAssessing Mathematical ontentKnowledge
Indicateourknowledgef eachword/phrasebelowbywriting 1, 2, 3, 4, or 5 infront f it.Thenumbersignifyhefollowingivestatements.
1. I'venever een theword/phrase.2. I've een theword/phrase,utIdon'tknowwhat t
means.3. I know the word/phrase as something o do
with..4. Ithink knowwhat tmeans nmath.5. I know he word/phrasen one or several of its
meanings,ncludinghemeaningormathematics.
UsingMeasures ndEquations
continuous supplementarynglesopposites acute riangleline equationlength fa segment equivalentquations
ray irrationalumbercentral ngleof a circle perfect ubecomplementaryngles absolute alueverticalngles segmentright riangle congruentegmentssolving nequation vertex fananglerational umber straight ngleperfect quare congruentnglesdiscrete obtuse rianglescientific otation solutionendpoint squarerootmidpoint realnumberangle cuberootright ngle
* The student'smathematical ontentknowledge* The student'smathematicalprocesses, such as reason-
ing, communicating,problemsolving, andmakingcon-
nections* The student's mathematicaldisposition, such as atti-
tudes,persistence,confidence,andcooperativeskills
Assessing MathematicalContent Knowledge
If you agree with the notion that words are labels for
concepts,thenyou will wantto use the ideas shown in fig-ure 1. I use this exercise as an informal pre- and post-assessment.At the beginningof a new unit or chapter and
againat theend), I give studentsa sheet similarto this one,with vocabulary erms for the unit listed.Thefirst time youuse theexercise, it is necessaryto go overthe five different
levels of work knowledge, but studentseasily understand
the idea that there arewords theyhave never heard of and
words thatthey know in several ways-and everythingin
between. (It is important o read the words aloud to the
class becausethereare some wordsthatstudentsrecognizewhen they hearthem but not when they see them.) Then,
have the studentswrite their best understanding f all the
words they rated as 4s or 5s so you can assess content
knowledge.This exercise is not used for a gradebutrather
as formativeassessment to give the teacheran idea of stu-
dents'understandings f the concepts before and after the
unit of instruction.
A second way of assessing students'contentknowledgeis the daily quiz sheet (Columbaand Dolgos 1993). Each
studentreceives a sheet (as shown in figure2) atthe begin-
ning of the week. Then, each day,eitheras studentsenter
class or as the closing activityfor the day, four problemsfrom a previousday's lesson or homeworkare given, and
students enter each problem (and solution) in the four
spaces for the day.The teacher can check these quicklyor
have a row gradercheck them. The sheets may be collect-
ed each day or at the end of the week, dependingon the
teacher'splanfor using the assessment nformation.
The thirdsuggestionfor formativeassessmentof content
knowledge is performanceassessment. Entirebooks have
been writtenon that subject;even thoughI cannot fullyexplain it in the context of this article,I would be remiss
not to mention t. Performance ssessmentsareassessments
"in which studentsdemonstraten a varietyof ways their
understanding f a topic or topics. These assessments are
judgedonpredeterminedriteria"ASCD 1996,59). Baron
(1990a, 1990b, and 1991) in MarzanoandKendall(1996)identifieda numberof characteristics f performanceasks.
Such tasks
FIGUREDailyQuizSheet
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94 TheClearingHouse November/December997
* aregrounded n real-worldcontexts,* involve sustainedwork and often take several days of
combined n-class and out-of-classtime,* deal with big ideas and majorconcepts within a disci-
pline,* presentnon-routine,open-ended,and loosely structured
problemsthat requirestudentsboth to define the prob-lem and to constructa strategy or solving it,
* requirestudents o determinewhat data areneeded,col-
lect the data,reportandportray hem,andanalyzethem
to discuss sourcesof error,and* necessitatethat studentsuse a varietyof skills for acquir-
ing informationand for communicating heirstrategies,
data,and conclusions. (93)
Figure3 is an exampleof a performance ask from a book
that is must readingfor anyone interested n performance
assessment, A Teacher's Guide to Performance-Based
FIGURESampleMathPerformance ask:Fast Food Math
BackgroundAlargenational amburgerast foodrestaurantchainprintedn advertisementhatread,"Sevenpercent f allAmericansat inourrestaurantsachday." hischainhas9,167restaurants.hereareapproximately50,000,000Americanitizens.
TaskYouareanemployee f a consumer dvocate roupstudyingruthnadvertising.ouhavebeen asked oconsider his advertisement.oyouthink t is proba-bly rueorprobablyalse?Your ask s to use logicalmathematicalrocedureso analyzethe claim hisadvertisementmade and writea summary f youropinion.AudienceThe audience oryour tudyandwrittenummarysthe person in charge of the consumeradvocategroup.
PurposeThepurpose fyouranalysisandwrittenummarysto show he use oflogicalmathematicalroceduresoexamine laimsmade nadvertising.
Procedure1. Review he assessment ist or ast foodmath.2. Rewritehe task nyourownwords.3. Makea listof all the informationhat is clearly
given.4. Make a list of what needs to be estimated,
assumed,or foundout.5. Make hose estimates nd/or ssumptions.6. Get other nformationhatyouneed.7. Useyourmath kills oanalyzehe advertisement.8. Write summaryfyour indingsoryourboss at
the consumer dvocacy roup.
Source:Educatorsn Pomperaug egional choolDistrict#15. 1996. A Teacher'sGuide o Performance-BasedLearn-
ing and Assessment. Middlebury,Conn.: PomperaugRegional School District#15. Copyright1996. Reprinted
Learning and Assessment (Educators 1996), written byteachers roma Connecticut chool district.I encourageall
readersof this issue to learn as much as they can aboutper-formance assessment. It is powerful-and it will change
your teachingforever!
Assessing Mathematical Processes
Manyof us would love to openup our students'heads so
we could see the wheels turningas they solve problems.Because that s notpossible,we have to find ways to get the
students to show us what they are doing. With some stu-
dents, that is fairly simple because they are extremely
metacognitivelyaware and don't mindlettingothersin on
their secrets. Other students,though, rarely "thinkabout
theirthinking"andso, of course,rarely et otherpeople in
on how they process (becausethey aren't even aware of it
themselves).The following ideas work with bothgroupsof
students(and everyone
inbetween);
wejust
have to be
more patientwith the lattergroup.The good news is that
once we begin askingadolescents to reflect on theirthink-
ing andproblem solving, it startshappening.
Interviewingstudents s a powerful way to tapinto their
processing. In the best of situations,teachers would have
time to interview students on an individual basis-at
length.In reality,we don't have the time to do this, so we
need to make time to (a) briefly interviewstudentson an
individualbasis and/or(b) interviewgroupsof students-
eitherbrieflyor at length.Long and Ben-Hur(1996) found
that the following phrasesor statementsare helpful when
interviewingstudentswho maynot be fully forthcomingor
expansive in their answers:
* I am nterestednyour hinking.* Pleasehelpme understand.uppose ouare heteacher nd
I amyourpupil.* I don't hink hat hisproblems easy.Sometimes getcon-
fused .. don'tyou?* SometimeswhenI havedifficultieswitha problem, break
it down ntosmall teps.Let'sdo thathereand indout ...[Theproblems modified.]
* I likeit whenyoutake he time o think.* I understandow,but ... (107)
Another marvelous idea for assessing students' mathe-
matical processes is shown in figure 4 (adapted from
Greenwood1996). By using this evaluationsheet, we givestudents an opportunity o think about their processing-and to give examples of what it is that they do. The first
time I use the sheet,I model whatit is I want students o do
and the specificity of examples I am looking for. Then, I
accept nothing ess than whatI know thatthey can do. Stu-
dents are sometimes resistantto this approachbecause it
involves a kind of thinkingthat some are not used to, butI
persevere,and within a shortperiodof time am able to getthe kind of formativeassessment data that is helpfulto me
in my teaching.
Regardlessof whetherwe areteaching seven-year-olds,
seventhgraders,or seventeen-year-olds,partof our task is
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Vol.71,No.2 FormativeAssessment in Math 95
FIGUREStudent/TeacherMathematicsEvaluation heet
What 'd What he Usethefollowingodeto mark Example(s)givemyself teacherwould yourselfn eachcriterion.
giveme
A = AlwaysM= Most of the time
O = OccasionallyS = SeldomN = Not at all
1. Icangiveclearand under-standablexplanationsfthethought rocessesIgothroughwhenIamsolvinga problem.
2. Icanuse thematerialsoshowthat hemathematicsIdomakes ense to me.
3. Whenevergetstuckon aproblem,canuse whatIknowogetunstuck.
4. Iamableto identifyrrorsinanswers,nthe use ofmathematics aterials,andinmy hinking.
5. When do a computation,don'talwaysneedpaperandpencil.
6. Whena strategy oesn't
work, tryanother neinsteadofgivingup.
7. Icanextend,orchange,aproblem yasking xtraquestions rposingdiffer-ent conditions.
8. Istudyandpractice eforetests andquizzes.
to teach them our expectations.If we expect students to
reflectontheir hinkingandproblem-solvingprocesses, henwe mustteach themto do so. Weneed to (a) modelourown
reflections, b) askthemabout heirreflections,and(c) givethem opportunities o reflect on their processes. In other
words, we can't give up just because they are resistant o
giving us information bouttheirmathematical rocessing.
Assessing MathematicalDispositions
Disposition is defined as "one's usual mood; tempera-ment,a habitual nclination, endency."Withteenagers, t is
not always easy to determine heirdispositionor tempera-ment regarding anything, including mathematics. Too
often, if theirreferencegrouphas decided that "school is
not cool," then it is mandatory hatthey use all theirbody
languageand facial expressions(andsometimeswords) toindicate their disdainfor our beloved subject.We mustn't
accept this judgmentat face value (no pun intended).The
ideas in this section will allow you to determineyour stu-
dents'mathematicaldispositions(sometimeswithouttheir
knowingit!).The first idea I want to offer came about almost by a
fluke. Another teacherand I were creatingan assessment
instrumentor some researchwe were conducting.We had
abouthalf a page left on theeight-page nstrument nddid-
n't want to waste thepaper,so we decided to pose theques-tion in figure 5 (McIntoshand Draper, n preparation). t
turnedout to be the best questionof the whole assessment!
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96 TheClearingHouse November/December997
FIGUREStudentSelf-Rating cale
20. Asa math tudentnthisclass, Iratemyself nthefollowingcale (putanXonthe scale whereyourateyourself).
< ----1---2----3 ----4----5-----6-...7-----9----0----->
Probablyhe Not oobad, Totallywesome!worstnthe not oogood. Maybehe bestclass. inthe school.
Iratedmyselfhiswaybecause
Considerusing this questionat the beginningof the year,and then several other times throughout he school year,to
get a sense of the changestakingplace.I have found thatadolescent studentsarewillingand able
to be more truthful when asked to write their thoughtsdown thanwhen asked to share hempublicly.For this rea-
son, I use learning ogs as often as possible to learnabout
students'dispositionstowardmathematics.The term earn-
ing log is not one thatI originated,butit is one that fits my
philosophyof how writinglooks in the mathematicsclass-
room. The first partof the term states the purposeof the
writing: earning.The secondpartconnotes aparticularor-
mat, thatis, runningcommentary.A log is not meant to be
a polished piece of writing,takenthroughdraft after draft.
Commander and Smith (1996) defined the purpose of
learning logs as "reflections on specific cognitive aspectsof learning... [emphasizing]the connection or personal
engagementwith academicskills andtechniques" 447).In the book WriteStarts:101 WritingPrompts or Math
(McIntoshandDraper1997), we offered a varietyof waysto use learning logs to assess students'attitudes,beliefs,and stereotypes about mathematics. The following are
some of the writingpromptssuggested:* What does a mathematician ook like?* My abilityto do math is ...* WhenI amin mathclass, I feel ...* Mathematics has good points and bad points. Here's
whatI mean ...* I study, I pay attention,I take notes, I read my math
book, but I still don't get math.Trueor false? Explain
youranswer.
Students' answers to questions such as these provide
insightto the teacher as she or he plans instruction. gnor-
ing students'dispositions towardmathematics s done at
teachers'-and students-peril.
Conclusion
We should not think that formativeassessment is some-
thing that is addedon to our alreadyfull curriculum.For-
mative assessment is partof good teaching.There should
be a seamlessness between instructionandassessment.The
word formative comes from the Latin word meaning
"shapeor form." Formativeassessment has as its purposeto shape upcoming instruction.If you use or modify the
ideas offeredin this article,you will find thatyourinstruc-
tion is more targetedand more effective. Then, you can
design more ideas of your own-and share them with as
manyother teachersas possible.
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Assessment:Readings from NCTM'sSchool-BasedJournals, 114-17.
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Commander,N. E, and B. D. Smith. 1996. Learning ogs: A tool for cog-nitive monitoring.Journal of Adolescent and Adult Literacy 39(6):446-53.
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