24 MatheMatics teaching in the Middle school ● Vol. 18, No. 1, Augus 2012
mMathematics lessons can take a variety o ormats. In this article, we discusslessons organized around complexmathematical tasks. These lessonsusually unold in three phases (Vande Walle et al. 2010). First, the task isintroduced to students. Second, stu-dents work on solving the task. Third,the teacher “orchestrates” a concluding whole-class discussion in which stu-dents are supported to make math-ematical connections between solutionsand to develop conceptual understand-ing o signicant mathematical ideas(Smith et al. 2009).
Complex tasks invite students togenerate multiple solutions and justiy
Consider our important elements o
setting up challenging mathematics
problems to support all students’ learning.
Kaa J. Jackson,Eily C. Saan,
Lynsey K. Gibbons,and Paul A. Cobb
their reasoning (Stein et al. 2000).Such tasks live up to their potentialonly i students engage productively in the task. Thereore, the rst phase,the task introduction, is crucial. Wedescribe how teachers can set up, or“launch” (Lappan et al. 2009), complextasks by leading introductory whole-class discussions in which they helpstudents engage in the task and learnsignicant math.
A task’s setup impacts both whatstudents and the teacher are ableto achieve during a lesson. When acomplex task is launched well andstudents are engaged, the many solu-tions generated can orm the basis o
the concluding whole-class discussion.Ideally, while students are solving thetask, the teacher circulates aroundthe room to plan or that concludingdiscussion. Students are more likely to learn rom a summary discussion i they have been able to engage produc-tively in the task.
Our work with middle-gradesmathematics teachers and our ownclassroom experiences indicate thatstudents requently struggle whenattempting to solve complex tasksbecause they do not understand various aspects well enough to getstarted ( Jackson et al. 2011). Teach-ers, in turn, oten respond to students’
26 MatheMatics teaching in the Middle school ● Vol. 18, No. 1, Augus 2012
struggles by repeating the task toindividuals or groups o students. Thistakes up a signicant amount o timeand can adversely aect the remaindero the lesson.
When teachers are spending timerelaunching a lesson, they are otenunaware o how other students aresolving the task. Thereore, they cannot plan well or the concluding whole-class discussion. In addition, tokeep the lesson moving, teachers otensuggest particular solution methodsto get students started. Such explicitsuggestions usually cause students tosolve the task in the same way, whichlessens the chance that a rich con-
cluding whole-class discussion willoccur. The end result is a diminishingo students’ opportunities to learnsignicant mathematics.
How can a teacher launch a com-plex task so that students are able toengage productively in solving it? Weanalyzed videorecordings o lessonsconducted by 132 middle-gradesmathematics teachers who usedStandards -based curricula to deter-
mine important aspects o eectivelaunches ( Jackson et al. 2011). Weidentied our aspects o launchesthat enabled the majority o studentsto engage in solving a complex task and to participate in a concluding whole-class discussion that ocused onconceptual understanding. We illus-trate these aspects by describing how a seventh-grade teacher, Smith (allnames are pseudonyms), eectively introduced a complex task, Dollars orDancing (see fg. 1), in his classroom.
Dollars or Dancing is a Standards -based task rom a unit on linear rela-tionships that emphasizes connectionsamong tables, graphs, and equations.Smith’s seventh graders had usedtables, graphs, and equations in priorlessons to solve problems involvinglinear relationships with y -interceptso zero. This lesson was their rst en-counter with a linear relationship and
nonzero y -intercepts. Smith’s goal inusing the Dollars or Dancing task wasto support his students’ understandingo the y -intercept as an initial value andslope as a constant rate o change.
Smith devoted eight minutes o an hourlong lesson to launching thistask. All his students were then ableto engage productively in nding asolution, and most contributed to theconcluding whole-class discussion in which they made connections among various strategies and representations.In doing so, they explained their rea-soning in ways that indicated con-ceptual understanding o key eatureso linear relationships (Jackson et al.
2011).
cRitical eleMents
We list our crucial aspects to keep inmind when setting up complex tasks tosupport all students’ learning.
1. Discuss the Key Contextual Features
Complex tasks oten involve real-worldscenarios that have been chosen tosupport the development o students’reasoning and communication aboutparticular mathematical ideas (Hiebertet al. 1997). Some students may struggle when starting because they areunamiliar with the context, or scenar-io, in which the task is grounded. It isthereore important that the teacherand students discuss any potentially
unamiliar eatures o the task. For
dollaRs FoR dancing
tee sudens a a scool ae aising dollas o e scool’s Valenine’s DayDance. All ee decide o aise ei oney by aving a dance aaon ine caeeia e eek beoe e eal dance. tey ill collec pledges o enube o ous a ey dance, and en ey ill give e oney o esuden council o ge a good DJ o e Valenine’s Day Dance.
• Rosalba’splanistoaskteacherstopledge $3.00 pe ou a se dances.
• Nathan’splanistoaskteacherstogive $5.00 plus $1.00 o evey ou e dances.
• James’splanistoaskteacherstogive $8.00 plus $0.50 o evey ou e dances.
Pr a. Ceae a leas ee dieen ays o so o o copae eaouns o oney a sudens can ean o ei plans i ey eac geone eace o pledge.
Pr B. Explain o e ouly pledge aoun is epesened in eac o you
ays o pa A.
Pr c. Fo eac o you ays in pa A, explain o e fxed aoun inNaan’s plan and in Jaes’s plan is epesened.
Pr d. Fo eac o e ays in pa A, so o you could fnd e aouno oney colleced by eac suden i ey could dance o 24 ous.
Pr e. wo as e bes plan? Jusiy you anse.
Source: Adaped o task 1.3 (“raising money”), Connected Mathematics Project 2 gade 7 book, Moving Straight Ahead: Linear Relationships (Lappan e al. 2009)
F. 1 te Dollas o Dancing ask alloed Si’s seven gades o expeience a
Vol. 18, No. 1, Augus 2012 ● MatheMatics teaching in the Middle school 27
example, key contextual eatures o Dollars or Dancing include under-standing that people organize dancemarathons to raise unds and thatdance marathons last a long time.
Unless these elements are discussed,students who are unamiliar with adance marathon might struggle early.
Smith anticipated that some o hisstudents were indeed unamiliar withthe dance-marathon scenario. Hebegan his launch by eliciting students’prior knowledge about dance mara-thons by projecting Internet imagesand asking students to discuss whatthey saw. Students replied, “Dance”and “Dance marathon.” He built on
students’ contributions to developa common way to describe dancemarathons as “groups o people whodance or a certain amount o time.”He then pressed students to explain why people might hold such an event. Their proposals were the oundationto explain that the task was a und-raiser to pay or a DJ or the school’sValentine’s Day Dance.
How can teachers help students
understand the key contextual eatureso a task? One idea is to ask studentsto imagine that they are participantsin the scenario and to share what they know about it. Another idea involvesprojecting images relevant to thescenario and asking students whatthey know about it. Images coupled with discussion are especially useulstrategies to support students whoare English language learners (ELLs)(Moschkovich 1999).
Teachers could also connect thetask scenario to a person, place, orevent that might be amiliar or o in-terest to students. For example, Smithpersonalized the Dollars or Dancing task or students by connecting it to aschool activity (the upcoming Valen-tine’s Day Dance) to generate studentinterest and to give students addition-al ways to contribute to the discussion(McDue et al. 2011).
2. Discuss the Key Mathematical Ideas
Our analysis o instruction indicated
that ocusing solely on context is notenough. It is also critical to discuss thekey mathematical ideas without hint-ing at particular methods or proceduresthat should be used to solve the task.For example, students were expectedto use tables, graphs, and equations inDollars or Dancing to represent theaccumulation o money over time inthe three dierent und-raising plans. To meet these expectations, studentsneeded to understand several ideas:
1. Money accumulates as a partici-pant continues to dance or agreater number o hours.
2. Dierent ways exist to accumulatemoney, such as starting with a xedamount or earning a xed amounto money per hour o dancing.
Unless students understood thisdistinction, they would be unlikely to
make connections among the dier-ent ways o accumulating money, theslope, and the y -intercept. It is alsoprobable that some students wouldstruggle both to create appropriate
tables, graphs, and equations, and laterto comprehend their peers’ represen-tations and explanations during the whole-class discussion.
Smith supported students’ under-standing o key mathematical ideas inthe Dance Marathon task by explicitly discussing the dierence between anup-ront amount and an hourly amount.
There are two ways that you canraise money in a dance marathon.
One way is to dance or a long time.I I give you $0.50 every hour, you’regoing to make a lot o money. Butthere’s another way that you couldraise money, and that is to ask or apledge. Not per hour, but just a [one-time] donation. We call that a dona-tion. And you might go up to yourteacher and say, “Can you give me$6.00 or being in the dance mara-thon?” That’s dierent. Can anybody
explain how that is dierent i I say,“Can you give me $6.00?” or instead“Can you give me $0.50 an hour?”
Smith called on students to explain thedistinction. Jasmine responded, “Eitherthey pay you up ront or . . . they con-tinue to pay you or however long youdance.” As students shared their ideas,Smith asked students to restate whatothers said (e.g., “Can you say what Jasmine said in your own words?”).He also praised students’ ideas andadopted students’ ways o describingthe distinction. Marisa oered, “One o them, you already start with it, and theother one, you have to kind o work or it to get more.” Smith responded,“Exactly. I like the way that’s worded.One o them you start with; you justhave it. The other one, you have to work or it to get the money.”
28 MatheMatics teaching in the Middle school ● Vol. 18, No. 1, Augus 2012
could describe in their own words thedistinction between the two ways toaccumulate money, Smith handed outthe task sheet and briefy explainedstudents’ responsibilities when work-
ing in their small groups. Studentsthen began the task. To develop his students’ under-
standing o key mathematical ideas,Smith ocused students’ attention onthe distinction between und-raisingstrategies, adopted students’ language,and asked multiple students to statethe distinction in their own words. We have also witnessed teachers ask-ing students to act out aspects o thetask to help develop an understand-
ing o key mathematical ideas. Smithmight have asked two students topretend to be Rosalba and Nathan. Then he might have asked “Rosalba”(hourly amount) and “Nathan” (up-ront amount) to explain how eachearned money. Other students wouldthen have been asked to explain thekey distinction between the two plans(Rosalba had to earn all o her money, whereas Nathan received some money
upront).
3. Develop Common Language to
Describe the Key Features
In the eective launches that weidentied, teachers did not simply talk to students about the key eatureso tasks but instead solicited input
from students. These teachers askedquestions that required more than yes or no responses, which helped theteacher determine the level o supportthat students needed to engage in thetask (Boaler 2002).
It was also critical or teachers tobuild on student contributions andboth support and press students todevelop common language to describethe key eatures o the task—contex-tual eatures, mathematical ideas, andany other language—that might beunamiliar or conusing. For example,Smith anticipated that the rst word
in part A, “create,” might be trouble-some or his students, especially ELLs (see fg. 1). He thereore askedstudents to explain the meaning o “create” using their own words during
the setup. Why is developing common language so important? Developing commonlanguage gives students a way tocommunicate with one another while working in small groups and partici-pating in the whole-class discussion. Teachers can use strategies similar tothose described by Smith to supportthe development o common language,such as highlighting particular ideas,adopting students’ language, asking
students to describe key aspects in theirown words, and asking students torestate what others have said (Chapin,O’Connor, and Anderson 2003).
4. Maintain the Cognitive Demand
To maintain the cognitive demand,or mathematical rigor, o the task (Stein et al. 2000), avoid suggest-ing a particular solution method tostudents. Doing so robs them o the
opportunity to develop mathematicalunderstanding as they generate theirown solution methods and representa-tions. Moreover, i students solve thetask in the same way, it is unlikely thatthe concluding whole-class discus-sion will present students with urtheropportunities to develop conceptualunderstanding o mathematical ideas.
With Dollars or Dancing, Smithcould have assisted his students by constructing a table, graph, or equa-tion. Although this would help allstudents get started, it would havealso reduced the cognitive demand o the task. Instead, Smith maintainedthe rigor by helping students under-stand important aspects o the task while leaving solution pathways open. This action allowed students to reasonabout signicant mathematical ideasboth while solving the task and whendiscussing it at the conclusion.
Planning coMPleX tasKs
Conducting high-quality launchesrequires considerable planning.Figure 2 provides a set o questionsthat teachers can ask themselves when
planning an eective launch.Smith’s launch was eectivebecause he had identied clear learn-ing goals or the particular lesson inlight o the mathematics standardso the state in which he taught. Healso anticipated contextual eatures,mathematical ideas, and languagecentral to the task that might not besel-evident to his students. Clearly,time is o the essence in classroominstruction. Thereore, teachers have
to make judgments about what war-rants attention in the launch o a task. These judgments must be based ona clear set o mathematical goals orinstruction and knowledge o whatmight be unamiliar to students.However, the time spent planning oran eective launch is worth it. Stu-dents are much more likely to be ableto get started solving a complex task,thereby enabling the teacher to attend
to students’ thinking and plan or aconcluding whole-class discussion. This, in turn, increases the chancesthat all students will be supported tolearn signicant mathematics as they solve and discuss the task.
