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Thinking through a Lesson: Successfully Implementing High-Level Tasks

Author(s): Margaret S. Smith, Victoria Bill and Elizabeth K. Hughes

Source: Mathematics Teaching in the Middle School , OCTOBER 2008, Vol. 14, No. 3 (OCTOBER 2008), pp. 132-138

Published by: National Council of Teachers of Mathematics

Stable URL: https://www.jstor.org/stable/41182652

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tation as a result of various classroom

factors. When this occurs, students

must apply previously learned rules

and procedures with no connection to

meaning or understanding, and the op-

portunities for thinking and reasoning

are lost. Why are such tasks so difficult

to implement in ways that maintain

the rigor of the activity? Stein and Kim

(2006, p. 11) contend that lessons based on high-level (i.e., cognitively chal-

lenging) tasks "are less intellectually

'controllable' from the teacher s point of

view." They argue that since procedures

for solving high-level tasks are often

not specified in advance, students must

draw on their relevant knowledge and

experiences to find a solution path.

Take, for example, the Bag of Marbles

task shown in figure 1. Using their

knowledge of fractions, ratios, and

percents, students can solve the task in

a number of different ways:

• Determine the fraction of each bag that is blue marbles, decide which

of the three fractions is largest,

then select the bag with the largest fraction of blue marbles

• Determine the fraction of each bag that is blue marbles, change each

fraction to a percent, then select

the bag with the largest percent of blue marbles

• Determine the unit rate of red

to blue marbles for each bag and

decide which bag has the fewest red marbles for every 1 blue marble

• Scale up the ratios representing each bag so that the number of

blue marbles in each bag is the same, then select the bag that has the fewest red marbles for the fixed

number of blue marbles

• Compare bags that have the same number of blue marbles, eliminate

the bag that has more red marbles,

and compare the remaining two

bags using one of the other methods • Determine the difference be-

tween the number of red and blue

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^H Ms. Rhee's mathematics class was studying statistics. She brought ia three bags containing red and blue marbles. The three bags were labeled às shown below:

A A A 75 red 40 red 100 red . . 25 blue 20 blue 25 blue ". 4

BagX Bag Y Bag Z Total = 100 marbles Total = 60 marbles Total = 125 marbles

Ms. Rhee shook each bag. She asked the class, "If you close your eyes, reach into a bag, and remove 1 marble, which bag would give you the best chance of picking a blue marble?"

Which bag would you choose?

Explain why this bag gives you the best chance of picking a blue marble. You may use the diagram above in your explanation.

marbles in each bag and select the

bag that has the smallest difference between red and blue (not correct)

The lack of a specific solution path is an important component of what makes this task worthwhile. It also

challenges teachers to understand the

wide range of methods that a student might use to solve a task and think about how the different methods are

related, as well as how to connect

students' diverse ways of thinking to

important disciplinary ideas.

One way to both control teaching with high-level tasks and promote suc- cess is through detailed planning prior to the lesson. The remainder of this

article focuses on TTLP: the Thinking Through a Lesson Protocol. TTLP is a process that is intended to further

the use of cognitively challenging tasks (Smith and Stein 1998). We begin by discussing the key features of the

TTLP, suggest ways in which it can be

used with collaborative lesson plan- ning, and conclude with a discussion

of the potential benefits of using it.

EXPLORING THE LESSON PLANNING PROTOCOL

The TTLP, shown in figure 2, provides a framework for developing lessons that use students' mathemati-

cal thinking as the critical ingredient in developing their understanding of key disciplinary ideas. As such, it

is intended to promote the type of careful and detailed planning that is

characteristic of Japanese lesson study

(Stigler and Hiebert 1999) by helping teachers anticipate what students will

do and generate questions teachers can ask that will promote student learning prior to a lesson being taught.

The TTLP is divided into three

sections: Part 1: Selecting and Set- ting Up a Mathematical Task, Part 2: Supporting Students' Exploration of the Task, and Part 3: Sharing and

Discussing the Task. Part 1 lays the

groundwork for subsequent planning

by asking the teacher to identify the

mathematical goals for the lesson and set expectations regarding how students will work. The mathemati-

cal ideas to be learned through work

Vol. 14, No. 3, October 2008 • MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 133

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on a specific task provide direction for all decision making during the lesson. The intent of the TTLP is

to help teachers keep "an eye on the

mathematical horizon" (Ball 1993)

and never lose sight of what they are trying to accomplish mathematically.

Part 2 focuses on monitoring students

as they explore the task (individu- ally or in small groups). Students are asked questions based on the solution method used to assess what they

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^H 1 ■ '

PART 1: SELECTING AND SETTING UP A MATHEMATICAL TASK

What are your mathematical goals for the lesson (i.e., what do you want students to know and understand about math- ematics as a result of this lesson)?

In what ways does the task build on students' previous knowl- edge, life experiences, and culture? What definitions, concepts, or ideas do students need to know to begin to work on the task? What questions will you ask to help students access their prior knowledge and relevant life and cultural experiences?

What are all the ways the task can be solved?

• Which of these methods do you think your students will use? • What misconceptions might students have? • What errors might students make?

What particular challenges might the task present to strug- gling students or students who are English Language Learners (ELL)? How will you address these challenges?

What are your expectations for students as they work on and complete this task?

• What resources or tools will students have to use in

their work that will give them entry into, and help them reason through, the task?

• How will the students work- independently, in small groups, or in pairs- to explore this task? How long will they work individually or in small groups or pairs? Will stu- dents be partnered in a specific way? If so, in what way?

• How will students record and report their work?

How will you introduce students to the activity so as to provide access to all students while maintaining the cognitive demands of the task? How will you ensure that students understand the context of the problem? What will you hear that lets you know students understand what the task is asking them to do?

PART 2: SUPPORTING STUDENTS9 EXPLORATION OF THE TASK

As students work independently or in small groups, what questions will you ask to -

• help a group get started or make progress on the task? • focus students' thinking on the key mathematical ideas

in the task?

• assess students' understanding of key mathematical ideas, problem-solving strategies, or the representations?

• advance students' understanding of the mathematical ideas?

• encourage all students to share their thinking with others or to assess their understanding of their peers' ideas?

How will you ensure that students remain engaged in the task?

• What assistance will you give or what questions will you ask a student (or group) who becomes quickly frustrated and requests more direction and guidance in solving the task?

• What will you do if a student (or group) finishes the task almost immediately? How will you extend the task so as to provide additional challenge?

• What will you do if a student (or group) focuses on non- mathematical aspects of the activity (e.g., spends most of his or her (or their) time making a poster of their work)?

PART 3: SHARING AND DISCUSSING THE TASK

How will you orchestrate the class discussion so that you accomplish your mathematical goals?

• Which solution paths do you want to have shared during the class discussion? In what order will the solutions be

presented? Why? • In what ways will the order in which solutions are

presented help develop students' understanding of the mathematical ideas that are the focus of your lesson?

• What specific questions will you ask so that students will -

1. make sense of the mathematical ideas that you want them to learn?

2. expand on, debate, and question the solutions being shared?

3. make connections among the different strategies that are presented?

4. look for patterns? 5. begin to form generalizations?

How will you ensure that, over time, each student has the oppor- tunity to share his or her thinking and reasoning with their peers?

What will you see or hear that lets you know that all students in the class understand the mathematical ideas that you intended for them to learn?

What will you do tomorrow that will build on this lesson?

134 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL • Vol. 14, No. 3, October 2008

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currently understand so as to move them toward the mathematical goal of the lesson. Part 3 focuses on orches-

trating a whole-group discussion of the task that uses the different solu-

tion strategies produced by students to

highlight the mathematical ideas that are the focus of the lesson.

USINGTHETTLPASA

TOOL FOR COLLABORATIVE PLANNING

Many teachers' first reaction to the

TTLP may be this: "It is overwhelm- ing; no one could use this to plan les-

sons every day'" It was never intended that a teacher would write out answers

to all these questions everyday. Rather,

teachers have used the TTLP periodi- cally (and collaboratively) to prepare lessons so that, over time, a repertoire

of carefully designed lessons grows. In addition, as teachers become more

familiar with the TTLP, they begin

to ask themselves questions from the

protocol as they plan lessons without

explicit reference to the protocol. This sentiment is echoed in the comment

made by one middle school teacher: "I follow this model when planning my lessons. Certainly not to the extent

of writing down this detailed lesson

plan, but in my mind I go through

its progression. Internalizing what it

stands for really makes you a better fa-

cilitator." Hence, the main purpose of

the TTLP is to change the way that teachers think about and plan lessons. In the remainder of this section, we

provide some suggestions on how you, the teacher, might use the TTLP as a tool to structure conversations with

colleagues about teaching.

Getting Started

The Bag of Marbles task (shown in fig. 1) is used to ground our discussion of lesson planning. This task would be classified as high level. Since no pre- dictable pathway is explicitly suggested or implied by the task, students must

access relevant knowledge and experi- ences, use them appropriately while working through the task, and explain

why they made a particular selection. Therefore, this task has the potential

to engage students in high-level think- ing and reasoning. However, it also has the greatest chance of declining during

implementation in ways that limit high-level thinking and reasoning (Henningsen and Stein 1997).

You and your colleagues may want

to select a high-level task from the cur-

riculum used in your school or find a

task from another source that is aligned

with your instructional goals (see Task Resources at the end of the article for

suggested sources of high-level tasks).

It is helpful to begin your collaborative

work by focusing on a subset of TTLP questions rather than attempting to

respond to all the questions in one sitting. Here are some suggestions on

how to begin collaborative planning.

Articulating the Goal for the Lesson

The first question in part 1 - What are your mathematical goals for the

lesson? - is a critical starting point for planning. Using a selected task, you can begin to discuss what you are

trying to accomplish through the use

of this particular task. The challenge is to be clear about what mathematical

ideas students are to learn and under-

stand from their work on the task, not

just what they will do. For example,

teachers implementing the Bag of Marbles task may want students to be able to determine that bag Y will give

the best chance of picking a blue mar- ble and to present a correct explana- tion why. Although this is a reasonable

expectation, it present no detail on what students understand about ratios,

the different comparisons that can be

made with a ratio (i.e., part to part,

part to whole, two different measures),

or the different ways that ratios can be

compared (e.g., scaling the parts up or down to a common amount, scaling

"Coming up with good questions before the lesson

helps me keep a high-level task at a high level, instead of pushing kids toward a particular solution path and giving them an opportunity to

practice procedures"

the whole up or down to a common amount, or converting a part-to-whole

fraction to a percent). By being clear on exactly what students will learn,

you will be better positioned to capi-

talize on opportunities to advance the mathematics in the lesson and make

decisions about what to emphasize and de-emphasize. Discussion with col- leagues will give you the opportunity

to broaden your view regarding the

mathematical potential of the task and the "residue" (Hiebert et al. 1997) that

is likely to remain after the task.

Anticipating Student Responses to the Task The third question in part 1 - What are all the ways the task can be solved? - invites teachers to move

beyond their own way of solving a problem and consider the correct and incorrect approaches that students are

likely to use. You and your colleagues

can brainstorm various approaches for solving the task (including wrong answers) and identify a subset of the solution methods that would be useful

in reaching the mathematical goals for the lesson. This helps make a

Vol. 14, No. 3, October 2008 • MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 135

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lesson more "intellectually control- lable" (Stein and Kim 2006) by encouraging you to think through the

possibilities in advance of the lesson and hence requiring fewer improvi-

sational moves during the lesson. If actual student work is available for the

task being discussed, it can help you

anticipate how students will proceed.

For example, reviewing the student work in figure 3 can provide insight into a range of approaches, such as

comparing fractions in figure 3d, finding and comparing percents in

figure 3b, or comparing part-to-part

ratios in figure 3g. Student work will also present opportunities to discuss incorrect or incomplete solutions such

as treating the ratio 1/3 as a fraction

in figure 3a, comparing differences

rather than finding a common basis

for comparison in figure 3f, and cor- rectly comparing x and z but failing

to then compare x and y in figure 3h. In addition, there should also be op- portunities to discuss which strategies might be most helpful in meeting the

goals for the lesson. Although it is impossible to predict everything that students might do, by working with

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Creating Questions That Assess and Advance Students' Thinking

The main point of part 2 of the

TTLP is to create questions to ask students that will help them focus on the mathematical ideas that are at

the heart of the lesson as they ex-

plore the task. The questions you ask

during instruction determine what students learn and understand about

mathematics. Several studies point to both the importance of asking good

questions during instruction and the difficulty that teachers have in doing

so (e.g., Weiss and Pasley 2004). You and your colleagues can

use the solutions you anticipated and create questions that can as- sess what students understand about

the problem (e.g., clarify what the student has done and what the stu-

dent understands) and help students advance toward the mathematical

goals of the lesson. Teachers can extend students beyond their current thinking by pressing them to extend what they know to a new situation or think about something they are not

currently thinking about. If student responses for the task are available,

you might generate assessing and

advancing questions for each antici- pated student response. Consider, for

example, the responses shown in fig- ure 3 to the Bag of Marbles problem. If you, as the teacher, approached

the student who produced response (c) during the lesson, you would notice that the student compared red marbles to blue marbles, reduced these ratios to unit rates (number of

red marbles to one blue marble), and

then wrote the whole numbers (3,

2, and 4). However, the student did not use these calculations to deter-

mine that in bag Y the number of red marbles was only twice the number of blue marbles, whereas in bag X and Z

136 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL • Vol. 14, No. 3, October 2008

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the number of red marbles were 3 and

4 times, respectively, the number of blue marbles. You might want to ask

the student who produced response

(c) a series of questions that will help

you assess what the student currently understands:

• What quantities did you compare and why?

• What did the numbers 3, 2, and 4

mean in terms of the problem? • How could the mathematical work

you are doing, making comparisons, help you answer the question?

Determining what a student under-

stands about the comparisons that he or she makes can open a window into

the student's thinking. Once you have a clear sense of how the student is

thinking about the task, you are better

positioned to ask questions that will advance his or her understanding and

help the student build a sound argu- ment based on the mathematical work.

POTENTIAL BENEFITS OF

USING THE TTLP

Over the last several years, the TTLP has been used by numerous elementary

and secondary teachers with vary-

ing levels of teaching experience who

wanted to implement high-level tasks in their classrooms. The cumulative

experiences of these teachers suggests that the TTLP can be a useful tool in

planning, teaching, and reflecting on

lessons and can lead to improved teach- ing. Several teachers have commented,

in particular, on the value of solving the

task in multiple ways before the lesson

begins and devising questions to ask

that are based on anticipated approach- es. For example, one teacher indicated,

"I often come up with great questions

because I am exploring the task deeper

and developing 'what if questions." Another participant suggested that pre-

paring questions in advance helps her support students without taking over

the challenging aspects of the problem for them:

Coming up with good questions

before the lesson helps me keep a

high-level task at a high level, instead

of pushing kids toward a particular

solution path and giving them an

opportunity to practice procedures.

When kids call me over and say they

don't know how to do something

(which they often do), it helps if I

have a ready-made response that

gives them structure to keep working

on the problem without doing it for

them. This way all kids have a point

of entry to the problem.

The TTLP has also been a useful tool

for beginning teachers. In an interview

about lesson planning conducted at the end of the first semester of her year-

long internship (and nearly six months after she first encountered the TTLP),

another preservice teacher offered the

following explanation about how the

TTLP had influenced her planning:

I may not have it sitting on my desk,

going point to point with it, but I

think: What are the misconceptions?

How am I going to organize work?

What are my questions? Those are

the three big things that I've taken

from the TTLP, and those are the

three big things that I think about

when planning a lesson. So, no, I'm

not matching it up point for point

but those three concepts are pretty

much in every lesson, essentially.

Although this teacher does not follow the TTLP in its entirety each time

she plans a lesson, she has taken key

aspects of the TTLP and made them part of her daily lesson planning.

CONCLUSION

The purpose of the Thinking Through a Lesson Protocol is to prompt teachers to think deeply

about a specific lesson that they will be teaching. The goal is to move beyond the structural components often associated with lesson plan-

ning to a deeper consideration of how to advance students' mathematical

understanding during the lesson. By shifting the emphasis from what the

teacher is doing to what students are thinking, the teacher will be better

positioned to help students make sense of mathematics. One mathe-

matics teacher summed up the poten- tial of the TTLP in this statement:

Sometimes it's very time-consuming,

trying to write these lesson plans, but

it's very helpful. It really helps the lesson

go a lot smoother and even not having

it front of me, I think it really helps me

focus my thinking, which then [it] kind

of helps me focus my students' thinking,

which helps us get to an objective and leads to a better lesson.

In addition to helping you create indi-

vidual lessons, the TTLP can also help you consider your teaching practice over time. As another teacher pointed out, "The usefulness of the TTLP is in

accepting that [your practice] evolves

over time. Growth occurs as the proto- col is continually revisited and as you reflect on successes and failures."

TASK RESOURCES

Bright, George W., Dargan Frierson Jr.,

James E. Tarr, and Cynthia Thomas.

Navigating through Probability in Grades

6-8. Reston, VA: National Council of

Teachers of Mathematics, 2003.

Bright, George W., Wallece Brewer, Kay

McClain, and Edward S. Mooney. Nav-

igating through Data Analysis in Grades

6-8. Reston, VA: National Council of

Teachers of Mathematics, 2003.

Bright, George W., Patricia Lamphere Jor-

dan, Carol Malloy, and Tad Watanabe.

Navigating through Measurement in

Grades 6-8. Reston, VA: National Coun-

cil of Teachers of Mathematics, 2005.

Vol. 14, No. 3, October 2008 • MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 137

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Brown, Catherine A., and Lynn V. Clark,

eds. Learningfrom NAEP: Professional De-

velopment Materials for Teachers of Math-

ematics. Reston, VA: National Council of

Teachers of Mathematics, 2006.

Friel, Susan, Sid Rachlin, and Dot Doyle.

Navigating through Algebra in Grades

6-8. Reston, VA: National Council of

Teachers of Mathematics, 2001.

Illuminations, illuminations.nctm.org/

Lessons.aspx. Parke, Carol S., Suzanne Lane, Edward

A. Silver, and Maria E. Magone. Using

Assessment to Improve Middle-Grades

Mathematics Teaching and Learning:

Suggested Activities Using QUASAR

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Art Johnson, Hannah Slovin, Carol

Malloy, and Ron Preston. Navigating

through Geometry in Grades 6-8. Res-

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of Mathematics, 2002.

Rachlin, Sid, Kathy Cramer, Connie Fins-

eth, Linda Cooper Foreman, Dorothy

Geary, Seth Leavitt, and Margaret

Schwan Smith. Navigating through

Number and Operations in Grades 6-8. Reston, VA: National Council of

Teachers of Mathematics, 2006.

REFERENCES

Ball, Deborah L. "With an Eye on the Mathematical Horizon: Dilemmas of

Teaching Elementary School Math-

ematics." The Elementary School Journal

93 (1993): 373-97. Boaler, Jo, and Karin Brodie. "The Im-

portance of Depth and Breadth in the

Analysis of Teaching: A Framework

for Analyzing Teacher Questions."

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Meeting of the North American Chapter

of the International Group for the Psy-

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773-80. Toronto, ON: PME, 2004.

^S^^toSet the Standards!/ Sample Tasks!

Problem Solving: Teach It, Learn It and Assess It

School CD-ROM for

grades 6-8 includes 120 problem-solving tasks for assessment, instruction and/or portfolio pieces. The CD is differentiated at three levels

of performance. Rubrics and

benchmark papers are provided.

Performance Tasks I Rubrics I Benchmark Papers

Henningsen, Marjorie, and Mary Kay Stein. "Mathematical Tasks and

Student Cognition: Classroom-Based

Factors that Support and Inhibit

High-Level Mathematical Thinking

and Reasoning." Journal for Research in

Mathematics Education 29 (November

1997): 524-49.

Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson,

Diana Wearne, Hanlie Murray, Alwyn

Olivier, and Piet Human. Making

Sense: Teaching and Learning Math-

ematics with Understanding. Ports-

mouth, NH: Heinemann, 1997.

Smith, Margaret Schwan, and Mary Kay

Stein. "Selecting and Creating Math- ematical Tasks: From Research to Prac-

tice. Mathematics Teaching in the Middle

School 3 (February 1998): 344-50.

Stein, Mary Kay, and Susanne Lane.

"Instructional Tasks and the Develop-

ment of Student Capacity to Think and

Reason: An Analysis of the Relationship

between Teaching and Learning in a Re-

form Mathematics Project." Educational

Research and Evaluation 2 (1996): 50-80.

Stein, Mary Kay, Barbara W. Grover,

and Marjorie Henningsen. "Building

Student Capacity for Mathematical

Thinking and Reasoning: An Analy- sis of Mathematical Tasks Used in

Reform Classrooms." American Edu-

cational Research Journal 33 (Summer

1996): 455-88.

Stein, Mary Kay, and Gooyeon Kim. "The Role of Mathematics Curriculum in

Large- Scale Urban Reform: An Anal-

ysis of Demands and Opportunities

for Teacher Learning." Paper presented

at the annual meeting of the American Educational Research Association, San

Francisco, California, April 2006.

Stigler, James W., and James Hiebert.

The Teaching Gap: Best Ideas from the

Worlds Teachers for Improving Educa- tion in the Classroom. New York: The

Free Press, 1999.

Weiss, Iris, and Joan D. Pasley. "What Is

High-Quality Instruction?" Educational

Leadership 61 (February 2004): 2Ф-28. •

138 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL • Vol. 14, No. 3, October 2008

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