When StudentsThrough tiered instruction, students atdifferent ends of the ability spectrumfind success in math class.

David Suarez

Iused to wonder why, despite myenthusiasm for teaching and mystudents' genuine interest inlearning, I was missing the markwith so many middle school math

students. However, after 1 put togetherVygotsky's (1986) concept of the zone ofproximal development with MihalyCsikszentmihalyi's perspective on howto create joyful concentration or "fiow"in leaming, 1 understood my classroomdynamics better. According to Csik-szentmihalyl (1990), enjoyment inleaming "appears at the boundarybetween boredom and anxiety, when thechallenges are just balanced with aperson's capacity to act" (p. 52). Recog-nizing this truth, I couldn't ignore theobvious. My underperforming studentswere either bored or overwhelmed.

When I began teaching integratedalgebra-geometr)' classes to 8th gradersat Jakarta International School inIndonesia, 1 decided to structure myclassroom so that students could choosetheir ov/n zone of proximal develop-ment, the leaming task that is just chal-lenging enough to be motivating.Jakarta International School is a private,international K-12 institution withapproximately 650 students in themiddle grades (6-8). My studentsranged from those who had beenrecommended for remedial mathcourses to those who had already

learned much of the upcoming year'smathematics curriculum. The questionof how to give each student in thisdiverse group the opportunity to growweighed on my mind.

As I struggled to select an appropriatecommon learning destination for agroup in which the starting points wereso different, I concluded that there wasno single appropriate end goal. Instead,in 2005, a fellow teacher and I devel-oped a tiered program of instructionthat enables students to study the samecontent at different levels of challenge.We began implementing this systemwith 8th grade math students in the2006-07 school year and are consid-ering using a similar approach in 8thgrade science.

Setting Up Tiered instructionWe found that the most helpfulapproach to tiered assessments was toorganize units of study into thematicunits, with specific skill outcomes desig-nated for each theme. For example, in athematic unit on graphing, skilloutcomes included converting betweengraphical, numerical, and symbolicrepresentations of data and analyzingfunctions and patterns. Establishingthese broad units enabled us to keep thenumber of traditional summative assess-ments to a manageable level byassessing at the end of each thematicunit rather than assessing each skillseparately.

Our next step was to di.stinguishamong foundational, intermediate, andadvanced levels of understanding foreach math skill. We designated eachlevel of mastery by a color. Studentschoose their color level for eachthematic unit and have the opportunityto vary their choices from unit to unit.

Green-level tasks meet the standardfor proficiency for 8th grade mathe-matics at Jakarta International School.

60 EDUCATIONAL LEADERSHIP/NOVEMBER 2007

Choose the Challenge

Blue-level tasks extend familiar skillsinto more complex work. To succeed atblue casks, students must be able torecognize the subtleties that make aproblem more complex and must besharp in the required skills. Black-levelchallenges are tbe most complex and areappropriate lor bighly advanced andmotivated studems. They require thecreative application and extension ofskills and sometimes require students to

carry out unfamiliar tasks. Figure 1presents examples of tasks at each levelof challenge.

The Power of Student ChoiceAt the outset of my algebra-geometrycourse, 1 explained the different levels oftasks and assessments and asked eachstudent to select the level of challengemost appropriate for his or ber indi-vidual learning, I encouraged students

to select the level of challenge thatwould help them maximize the speedand quality of their learning. Drawingon Jensens (1998) work on bow stressaffects learning, 1 explained thatchoosing tasks that were too hard or tooeasy would lead to less than ideal stresslevels: Tasks that were just challengingenough would make learning interesiingbut not ovei'whelming.

My underperformingstudents were eitherbored or overwhelmed.

Consistent with Glasser's (1986)choice theory, empowering students toselect tbeir own level gives three majorbenefits:

• Students tind choices motivating—olten the key to achievement ii r middleschoolers,

• Students benefit from the opportu-nity to make decisions. Learning toreflect on personal learning and adjusttasks accordingly is a great skill formiddle school students.

• Students can't conclude that agrouping decision made on their behalfis untair or inappropriate.

My students consistently madeappropriate choices and enthusiasticallyaccepted this responsibility. Onestudent. Rutb, wrote.

Being asked to choose from three difterenilevels of difficulty has given me morechoices and opponuniLies to challengemyself. . . . I feel that I have more "say" inthe itve! of math I am learning.

A S S O C I A T I O N T O R S U P L R V I S I O N A N D C U R R I C U L U M 61

How It Works in the ClassroomA typical day in my classroom beginswith a brief period of whole-classinstruction intended to illuminate thelesson's essential understandings.Students then select practice assign-ments at the level of challenge theydesire. They work on these assignmentsfor approximately 40 minutes in classand complete them at home. Studentscan choose a different level every day ifthey want, and their methods of oper-ating vary. Some learners like tocomplete the same challenge level all thetime, others switch often between levels.

In-class practice is essential because itgives students the opportunity to workwith others who are interested in tack-ling similar challenges—and gives mean opportunity to help students. Moststudents enjoy working with others whoare attempting similar problems. There'susually a buzz in the air as students seekout others, either as partners in problemsolving or as helpers if a student isstuck. 1 seat students at heterogeneouslyarranged tables but let them move

Student performanceacross the gradelevel has increased.

around the room as needed during indi-vidual practice time.

l-igure 2 shows three different levels ofassignments that I provided my studentsafter a lesson on triangle propenies. Thislesson was one of several in a thematicunit on angle relationships. At the unit'sconclusion, students selected a summa-tive assessment linked lo angle relation-ships at a challenge level of their choice.

I allow students to look at all of theassessments, compare them, and maketheir choice: They can even start oneassessment, quit, and switch to adifferent level test if time allows. Astudent who does poorly on a higher-level assessment can go back and takethe green-level assessment to demon-strate at least grade-level proficiency.

The ResultsHigher Achievement.. .Our teachers implementing tiered mathinstruction have been extremely pleasedwith the results so far. Students areperforming at higher levels of achieve-ment, are more motivated, and areassuming more responsibility for theirleaming. Whal has surprised us is that,even though we have used the founda-tional level of performance in 8th grademathematics as the "starting" green level,a level that no student is compelled toexceed, student perfomiance across thegrade level has increased.

Students at our school seem to strivefor, and achieve, an A-/B+ level of excel-lence no matter what assessment levelthey select. It follows, then, that if theyare scoring at such a level on harder tests,their achievement has risen. By midwaythrough the 2006-07 school year,teachers had asked 8th graders to make atotal of 883 choices among leveLs of mathassessments, and students most oftenchose harder-than-basic tests. Studentsselected green-level tests, which are

FIGURE 1. Sample Student Tasks at Different Challenge Levels

LessonTopics

Pro

blem

sol

ving

with

lin

ear e

quat

ions

Und

erst

andi

ng s

lope

Green-level tasks(foundational)

The difference in the ages of twopeople is 8 years. The older personis 3 times the age of the younger.How old is each?

Find the slope of the line passingthrough the following pair of points:(-4, 6) and (-3, 2).

Blue-level tasks(intermediate}

The length of a rectangle is 3 lessthan half the width. If the perimeteris 18, find the length and width.

Find a so that the line connectingthe points (-2, -3) and (2, 5) isparallel to the line connecting thepoints (6,a) and (0, -4).

Black-level tasks(advanced)

When asked for the time, aproblem-posing professor said, "Iffrom the present time, you subtractone-sixth of the time from now untilnoon tomorrow, you get exactlyone-third of the time from noon untilnow." What time was it?

If a > 1, what must be true aboutb so that the line passing throughthe points (a, b) and (1, -3) has anegative slope?

62 EDUCATUINAL LtADLR.SinP/NDVtMbLK 2007

similar to the whole-group assessmentswe used before introducing choice, only33 percent of the time. They chose blue-level assessments (above the proficiencystandard) 59 percent of the time, andblack-level tests 8 percent of the time.Thus, students are now tackling greaterchallenges than in the past.

At the same time, test scores areholding steady compared with previousyears, indicating that overall achieve-ment has risen. In 2005-06, students in8th grade algebra-geometry achieved anaverage score of 90 percent correct onthe whole-group assessments they alltook throughout tbe year (which were atthe green level of difliculty). During thefirst year of tiered instruction, studentsin this class achieved average scoreswithin a few points of 90 percentcorrect on tbe green-, blue-, and black-level assessments they took.

Before we launched tiered instruction,students at the beginning end of thereadiness spectrum tended to bring aclass's average math test scores down.Now, instead of bringing average scoresdown on a whole-group assessment,students at the beginning end of thereadiness spectrum score on green-levelassessments at levels comparable tothose of students taking the harderblue- and black-level tests. 1 have beenthrilled to see students at this "greenlevel" improve their performance.

. . . And Eager LearnersStudents consistently showed increasedmotivation once I gave them choices. Ifstudents believe themselves to be belowaverage, they will generally performbelow average on an assessmentdesigned for the entire class. On theother hand, students who believe thatassessments were designed with theirreadiness level in mind will expectthemselves to be successful. Tieredlearning fuels a positive self-fulfillingprophecy.

Students learn a lot about themselves as theygrapple with questions about what is best forthem and move forward with new insights.

Positive PerspectivesBoth students who have a history ofstellar achievement in math class andstudents who start with more basic skillsleel comfonable with the tiered system.Gabi, who tends to select green-levelassignments, commented, "I like havingchoices because you can decide whetheryou are ready for a harder challenge ornot." Other students are enthusiasticallytackling unprecedented levels ofachievement. After taking a black-levelassessment, Wa-Lee exclaimed, grinning,"That was hard!" When I asked Johanneshow he was feeling about an upcomingblack-level test, he replied, "Excited!"

At the conclusion of each unit, we askstudents to reflect on how difficult oreasy the work they picked was for them.Overwhelmingly, students reportedfeeling appropriately challenged. On 88

percent of the written reflections that 8thgraders completed, students reportedfeeling appropriately pushed "toward thegoal of maximizing their learning." Ononly 7 percent of these reflections didstudents label the assessment that theyhad selected as "too simple" for them,and on only 6 percent did they labeltheir choice "too challenging."

Offering tiered choices allowsstudents to modify future decisions if, inhindsight, they view an assessment theyhave selected as too simple or too chal-lenging. With this arrangement, onestudent's growth and success in mathneed not come at the price of another'schance for the same. In fact, a very posi-tive classroom culture has developed.Peer pressure now wields a positiveinfluence as students take pride Inconfronting challenges and at times

A S S O C I A T I O N [ - O R S U P E R V I S I O N A N D C U R R I C U L U M 63

choosing a higher level.My students learn a lot about them-

selves as they grapple with questionsabout what is best for them and moveforward with new insights. Studentresponses to the prompt "How did youselect your challenge level? Are yousatisfied with your choice?" arerevealing. Vishali noted,

IBlue-level work] is what I am comfort-able wiih. 1 know I am capable of blue, Iam satisfied with my choice because ilearned and understood many newthings. I know that if I had chosen black,then I would have been stuck in chaos.

Tanisha commented,

The three-choice color system helpsimprove leaming because it gives you thefeeling that no one is forcing you to dosomething that you might find toostressful. It also gives you a better idea ofhow to be independent and not haveeverything be decided for you.

Parents' reactions have also been veryfavorable. Parents of advanced studentsfinally fee! their children are being chal-lenged in class, whereas the parenl of astudent at lhe other end of the readinesscontinuum remarked, "This is the firsttime my child feels successlul m math."Another mother enthused.

This is making my daughter think abouther learning and it gives her a chance topractice decision making. This is exactlywhat kids should be doing in middleschool.

A Work in ProgressMy colleagues and I recognize that ourefforts are a work in progress. But ourtiered approach to leaming and assess-ment is positively affecting studentachievement, and students and parentsbolh prefer il to our previouscurriculum. I do have a few words ofcaution, encouragement, and supponfor schools considering a similar effort.'

First, the process of developing tieredassessments and differentiating instruc-

tion gets easier over time. If you areapprehensive, move ahead and don'tparalyze yourself with worry Second,keep your eyes open for challengingmath problems. There is no shortage offoundational-level problems in tradi-tional textbooks. We have, however,found that there is a short supply of

high-level problem-solving tasks.Supplementary challenge handoutsavailable in the teacher support mate-rials accompanying textbooks are goodplaces to Stan. I also keep my eyes openfor challenging problems in mathe-matics books and Web sites.^

Finally, develop a grading system that

My "Aha! " MomentM. Kathleen Heid, Distinguished Professor of Curriculum and Instruction,Penn State University, University Park, Pennsylvania.

I have had several early "aha" moments in mathematics, at different times andin different contexts. As a 10-year old, I was riding with my family across theNorthern Tier of Pennsylvania, returning from a family vacation trip in NewEngland. The trip was a long one, so I occupied myself withseveral mental games. I would try to predict when wewould reach certain landmarks on the trip, record thepredictions, and then check them when we arrived.As the trip progressed, 1 became increasingly accu-rate in my predictions. Next, I decided to start a listof numbers, doubling the last one for each one Iwrote. I started noticing patterns and predicting thesequence of digits in decimal places of consecutivenumbers. Both times, 1 was surprised and pleased tonotice the power in numbers (in the accuracy of predic-tion and in the regularity in digit patterns),

Another instance of a growing excitement about mathematics occurred thesummer after 8th grade. To get ready for high school, I purchased an introduc-tory book on physics. I was delighted to find that there were formulas thatdescribed physical action and that I could use those formulas to figure outquantities that were not given in problems. Key to excitement for young peopleis the ownership they can feel when their discoveries^no matter how smal l -are of their own making.

Copyright © Kathleen Heid

64 EDUCATIONAL LLADE^RSIM p/NovtMi(fiR 2007

FIGURE 2. Tiered Work for a Lesson on Triangle Properties

Green-level task(foundational}

Blue-level task(intermediate)

Black-level task(advanced)

Find the value of

(2x+ 15)°

In A ABC. the measure ofthree times that of

. and the measure ofz. S is twice the sum of the

measures of ^A andFind the measure

of each angle.

Source: Green-Sevel task is from Houghton Mifflin Unified Mathematics Book 1 by G. Rising,

1991, Atlanta: Houghton-Mifflin. Copyright 1991 by Houghton-Mifflin. Used with permissran.

Black-level task is from Challenging Problems in Geometry, by A. Posamentier and

C. Salkind, 1988. Nev iYark: Dover. Copyright 1988 by Dover. Used vi/ith permission.

In any A A^. f and Dare interior pointsof /AC and BC. respectively.

^Fbisects ^^CAD. and SFbisects ^-CBE.Prove m ^AEB + m ^ADB = 2m^ AFB

Is compatible with the differentiationpractices you implement and be trans-parent about it with students andparents. We experimented during ourfirst year of tiered instruction withweighting grades based on the level ofchallenge the student selected. Thisschool year, however, the JakartaInternational School is attempting"standards-based reporting." We reportstudent performance against individuallearning goals (rather than reportingone overall course grade) and reportboth the level of difficulty selected byihe student for each learning goal (as aperformance level) and the accuracywith which the student demonstratesmastery (as a letter grade). For example,a student might earn an A, B, or Con a task at any of the levels for anyparticular goal.

We continue to contemplate issuessuch as when and how much to guidestudents in their decision making, howlo improve differentiation practicesduring instruction, and how to handle asituation in which the green level of

Tiered learning fuelsa positive self-fulfilling prophecy.

challenge is beyond a student's readinesslevel. Fm excited at these opportunitiesfor continued exploration into tieredinstruction and assessment. The joumeyso far has left me feeling closer than everbefore to my goal of meeting students'needs as math learners. SI

' For a more detailed explanation of howto implement a tiered instructional programin math, see Chapter 11 of Making the Differ-ence: Differentiation in International Schools(Powell &r Kusuma-Powell, 2007) or visitmy blog at www. challengebychoice.word press, com/

^ I have found the following resourceshelpful as sources of higher-level problems:Challenffng Problems in Algebra and Chal-ienging Problems in Geometry, (Posamentier &Salkind, 1988). The MathCounts School Hand-books, available at www.mathcounts.org.

Balanced Assessments in Mathematics,available at http://balancedassessment.concord.org

ReferencesCsikszencmihalyi, M. (1990). Flow: The

psychology of optimal experience. New York:Harper and Row.

Glasser, W (1986). Choice theory in the class-room. New York: Harper and Row.

Jensen, E. (1998). Teaching with the brain inmind. Alexandria, VA: ASCD.

Posamentier, A., &r Salkind, C. (1988).Challenging problems in algebra. New York:Dover.

Posamentier, A-, & Salktnd, C. (1988).Challen^ng problems in geometry. NewYork: Dover.

Powell, W, &r Kusuma-Powell, O. (2007).Making the difference: Differentiation ininternational schools. Washington, DC:Overseas Schools Advisory Council, U.S.Department of State.

Vygoisky, L. (1986). Thought and language.(A. Kozulin, Ed.). Cambridge, MA: MITPress.

David Suarez teaches 8th grade mathand science and heads the math depart-ment at the Jakarta International School,P 0. Box 1078-JKS, Jakarta 12010Indonesia; dsuarez@jisedu.or.id.

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