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







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







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!







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.

REFERENCES

Association for Supervision and CurriculumDevelopment (ASCD).

Developing performanceassessments. Facilitator'sguide. Alexandria,Va.:ASCD.

Clarke, D., and L. Wilson. 1996. Valuingwhat we see. In Emphasis on

Assessment:Readings from NCTM'sSchool-BasedJournals, 114-17.

Reston, Va.:National Council of Teachersof Mathematics.Firstpub-lished in MathematicsTeacher Oct. 1994).

Columba,L., and K. Dolgos. 1993. Tips for beginners:Daily-quizsheet.MathematicsTeacher86: 378-79.

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.

Educatorsn Connecticut'sPomperaugRegionalSchool District 15. 1996.A teachers' guide to performance-basedlearning and assessment.

Reston,Va.:AssociationforSupervisionand CurriculumDevelopment.Greenwood,J. J. 1996. On the natureof teaching and assessing mathe-

maticalpower and mathematical hinking.InEmphasison assessment:

Readings rom NCTM'sschool-basedjournals, 106-108. Reston, Va.:National Council of Teachersof Mathematics.Firstpublished n Arith-metic Teacher Nov. 1993).

Long, M. J., andM. Ben-Hur.1996. Informing earning through he clin-ical interview.In Emphasison 'assessment':Readingsfrom NCTM'sschool-based journals, 106-108. Reston, Va.: National Council ofTeachersof Mathematics.Firstpublished n ArithmeticTeacher (Feb.1991).

Marzano,R. J., andJ. S. Kendall. 1996.A comprehensive uide to design-

ing standards-baseddistricts,schools,and classrooms.Alexandria,Va.:

Association for Supervisionand CurriculumDevelopment.McIntosh,M.E., and R. J.Draper.1997.Write tarts:101 writingprompts

for math.PaloAlto, Calif:. Dale Seymour.- "There,I finally said it": Middle School Students'Self-Ratings.

Manuscriptn preparation.National Councilof Teachersof Mathematics NCTM). 1995. Assessment

standardsor school mathematics.Reston,Va.:NCTM.

Pregent,R. 1994. Chartingyour course: How to prepare to teach more

effectively,Madison,Wis.:MagnaPublications.

Wilcox,S. K., andR. S. Zielinski. 1997.Using the assessmentof students'

learning o reshape eaching.MathematicsTeacher90(3): 223-29.





1. Formative Assessment in Mathematics

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