mathematics for ell students workshop 2 presentation

Teaching High-Level Mathematics to English Language Learners

in the Middle Grades

1135 Tremont Street, Suite 490Boston MA 02120

© Copyright 2009 Center for Collaborative Education/Turning Points

Workshop 2

Teaching High-Level Mathematics to English Language Learners in the Middle Grades was developed by Turning Points, a project at the Center for Collaborative Education in Boston, MA. This tool is part of the Mathematics Improvement Toolkit, a project of the National Forum to Accelerate Middle Grades Reform, and was supported by the U.S. Department of Education’s Comprehensive School Reform Initiative, grant #S332B060005. Opinions expressed are those of the authors and are not necessarily those of the Department.

Developed byTurning Points, a project of

the Center for Collaborative Education

WELCOME!

WRITE

Write your name on a name-tag with the markers on the table (if you don’t already know everyone in the workshop).

TAKE

Please take a participant’s packet if you haven’t already taken one.

BEST PRACTICE: VISUAL CUESEasy-to-identify, visual cues •call attention to key points and critical classroom routines,

•reduce the language load,

•create predictable and easy-to-follow routines, and

•allow English language learners to participate quickly and actively in lesson.

Workshop 2 Agenda

Warm-up/Checking In

Big Ideas of Workshop 2 –

Mathematical Tasks and Cognitive Demand

Comparing the Cognitive Demands of Two Tasks:

Linguistic, mathematical and cultural

Doing Mathematics

Classroom Video #1

Doing Mathematics: Finding the Best Box

Reflecting on Doing Mathematics

Keeping Classroom Activities at a High Level of Cognitive Demand

Action Plans: Next Steps in Your Own Classroom

Closing Circle

2-1

Teaching High-Level Mathematics to English Language Learners in the Middle Grades


In workshop 1, you learned:•who the English language learners in schools are today•how to recognize some of the challenges faced by English language learners in learning high-level mathematics•how to support English language learners in learning the language of mathematics•how to ensure the active engagement of English language learners in developing the mathematical reasoning essential to mastering high-level mathematics

In workshop 2, you will learn:•how to distinguish high-level from low-level mathematical tasks in the middle grades•how to create and support high-level mathematical activities that engage English language learners in the middle grades

2-2

Warm-Up / Checking In

Read HANDOUT 2-3: Best Practices Modeled in Workshop 1

WRITE on HANDOUT 2-4: one practice from the list that you have added to your personal toolkit.

Explain how it has changed your teaching and

how it has helped the English language learners

in your classroom to learn high-level mathematics.

Warm-up/Checking In2-3, 2-4

Sample questionsWhat surprised you? What did you learn from the change you made? What questions does the change you made raise for you about teaching English language learners?

Warm-up/Checking In (continued)

Round 1

I used ... with my students because

SPEAK

I learned that my students...

RESPONDRETELL

You said you tried …because...

What did you learn fromthe change?

FOLLOW-UP QUESTION

Speak, Retell, Follow-up Question, Respond

Round 2: reverse roles and repeat the process

2-5

PAIR:Best Practice:•Ensures both parties are involved throughout the conversation. No one can dominate.

•RETELL provides a reason for listening as well as a check for understanding

•FOLLOW-UP supports a deepening of both parties’ understandings.

•RESPOND gives the first speaker a chance to clarify her own thinking.

Warm-up/Checking In (continued)

Reflecting on your experience

Take some notes for yourself.

Why did you choose this particular strategy to try out?

What did you learn through sharing with a partner?

2-6

Share

Introducing the “Big Ideas” of Workshop 2

Mathematical Tasks, Cognitive Demand and High-level Mathematics in the Middle Grades

This workshop is partially based on research by the Quasar Project, at the University of Pittsburgh. This project:

• Conducted five years of research in urban middle schools,

• Found that mathematically rich, cognitively demanding tasks, implemented appropriately in classrooms are the key to success for urban students,

• Developed a teaching approach to ensure that all students are engaged in problem solving, mathematical thinking, mathematical communication. This is what we call “high-level mathematics” or “doing mathematics.”

Quasar Project findings about teaching high-level math to urban middle school students

Research-based strategies that support English language learners to develop critical thinking, academic language, communication skills

A coherent approach to teaching high-level math to English language learners

High-Level Mathematics for English Language Learners

Research on Learning High-Level Math

Research on English Language Learners

This Project

“Big Idea” of the Quasar Project:

High Levels of Cognitive Demand Lead to Substantial Learning Gains

“… students who performed best … were in classrooms in which tasks were … set up and implemented at high levels of cognitive demand … For these students, having the opportunity to work on challenging tasks in a supportive classroom environment translated into substantial learning gains on an instrument designed to measure student thinking, reasoning, problem solving and communication.”* * A Quote from: Implementing Standards-Based Mathematics Instruction: A Casebook

for Professional Development. Teachers College Press, 2000, by M. K. Stein, M. S. Smith, M.A. Henningson and E. A. Silver.

2-7

Pair with someone and discuss the quote using the Final Word Protocol

Best Practice:

FINAL WORD

•efficient way to discuss a reading

•provides controlled choice

•makes sure all voices are heard

•ensures in-depth discussion of issues important to group

•provides entry point into discussion for those who have not been able to read or absorb entire document

•English language learners can choose text that they can understand to discuss w. group

Final Word Protocol

• Pair with another person.

• Each of you choose a brief excerpt from the

quote: a phrase or passage to discuss. Decide who will speak first.

• Person#1 reads the excerpt and explains what it means to them and why it is significant. Person #1 has 1 minute to speak. If they finish before the minute is up, person #2 waits until the minute is over before responding.

• Person #2 retells what they heard person #1 say and adds a comment.

• Person #1 has the last comment, the Final Word, taking into account all that has been said.

• Switch roles and repeat the protocol starting with Person #2’s phrase.

SHARE

2-7, 2-8

#1

#2

#1

The cognitive demand of a task is the sum total of what a student needs to know, understand and be able to do in order to solve a problem or complete a task successfully.

What is Cognitive Demand?

The level of cognitive demand depends not only on the task, but also on the prior knowledge of the students.

2-7

The cognitive demand of a task is the sum total of what a student needs to know, understand and be able to do in order to solve a problem or complete a task successfully.

The level of cognitive demand depends not only on the task, but also on the prior knowledge of the students.

What is Cognitive Demand?

2-7

•Person #1 speaks by reading the first sentence and explaining what s/he thinks it means.

•Person #2 retells what person #1 said.

•Person #2 asks a follow-up question.

•Person #1 responds.

•Switch Roles. Person #2 starts by reading the second sentence.

SHARE

Discuss these statements with a partner using

Speak, Retell, Follow-up Question, Respond

Comparing the Cognitive Demands of Two Tasks

Consider the cognitive demands of the following two tasks for English language learners (HANDOUT 2-9)

Task 1. Find the surface area and volume of a rectangular prism that measures 2” x 4” x 24”.

Task 2. Out of This World Candies plans to sell Starburst candies in a new package containing 24 individually wrapped Starbursts. Your challenge is to find the dimensions of the least expensive box that can hold exactly 24 Starbursts. Each wrapped Starburst has a square shape that measures 2 cm on a side and 1 cm high.

Write a report including the dimensions for all the possible boxes we can use to package 24 Starbursts, the dimensions of the least expensive box, and explain how you know your answers are correct.2 cm

2 cm

1 cm

2-9

What are the cognitive demands of Task 1 for English language learners?

Task 1. Find the surface area and volume of a rectangular prism that measures 2” x 4” x 24”.

Think individually about the cognitive demands of Task 1. Consider the linguistic, mathematical and cultural demands separately.

• Write your answers for linguistic demands in the first column on Handout 2-10.

• Write your answers for mathematical demands in the first column on Handout 2-11.

• Write your answers for cultural demands in the first column on Handout 2-12.

2-10, 2-11, 2-12

Jigsaw Activity for Task 1

Divide into three (3) groups - linguistic, mathematical and cultural - for a jigsaw activity.

Each group will become an “expert” on their section.

Share your answers about Task 1, one at a time, with your own “expert” group. Come to consensus on the best responses:

•consider everyone’s responses respectfully,

•ask each other questions,•decide which responses are the

best, and•each member of the group

write the group’s responses on HANDOUT 2-13..

2-10, 2-11, 2-12, 2-13

Jigsaw Activity for Task 1 (continued)

Reassemble into groups of three people, with one linguistic expert, one math expert, and one cultural expert in each group. Each person needs to share your expert group’s responses with the other two.

Ask questions and make comments. Make sure that everyone becomes an expert on all three questions!

2-13

What are the cognitive demands of Task 2 for English language learners?

Think about the cognitive demands of Task 2 individually.

Write your answers for Linguistic, Mathematical and Cultural demands in the second columns of HANDOUTS 2-10, 2-11 and 2-12.

Task 2. Out of This World Candies plans to sell Starburst candies in a new package containing 24 individually wrapped Starbursts. Your challenge is to find the dimensions of the least expensive box that can hold exactly 24 Starbursts. Each wrapped Starburst has a square shape, that measures 2 cm on a side and 1 cm high.

Write a report including the dimensions for all the possible boxes we can use to package 24 Starbursts, the dimensions of the least expensive box for us to make, and explain how you know your answers are correct.

2 cm

2 cm

1 cm

2-10, 2-11, 2-12

Jigsaw Activity for Task 2

Return to your three expert groups - linguistic, mathematical and cultural - for a jigsaw activity.

Each group will become an “expert” on their section.

Share your answers about Task 2, one at a time, with your own “expert” group. Come to consensus on the best responses:

•consider everyone’s responses respectfully,

•ask each other questions,•decide which responses are the

best, and•each member of the group

write the group’s responses on HANDOUT 2-14.

2-10, 2-11, 2-12, 2-14

Jigsaw Activity for Task 2 (continued)

Reassemble into groups of three people, with one linguistic expert, one math expert, and one cultural expert in each group. Each person needs to share your expert group’s responses with the other two.

Ask questions and make comments. Make sure that everyone becomes an expert on all three questions!

2-14

Recall the Key Finding of the Quasar Project

Answer this question individually using Handout 2-15.

“… students who performed best … were in classrooms in which tasks were … set up and implemented at high levels of cognitive demand … For these students, having the opportunity to work on challenging tasks in a supportive classroom environment translated into substantial learning gains on an instrument designed to measure student

thinking, reasoning, problem solving and communication.”

2-15

If you want English language learners to engage in high-level mathematics and become successful mathematical thinkers, reasoners, problem solvers and communicators, would you use Task 1 or Task 2? Explain why.

Now let’s take a poll. How many chose

Task 1? _________ Task 2? __________

Doing Mathematics!

Low-level math

High-level Math

Mathematical tasks can be grouped according to their levels of Levels of

Cognitive Demand

• Low level tasks such as Task 1 involve memorizing facts and following procedures

• Intermediate level tasks involve using procedures and concepts

• The highest level tasks such as Task 2 involve doing mathematics, that is, solving novel and challenging problems

Doing Mathematics

When students are engaged in tasks at the highest level, we call this “Doing Mathematics.”

This will be the focus of the rest of this workshop.

We will begin with a thorough exploration of Task 2

Doing Mathematics

What kind of package does it come in?

How do you think candy makers decide what kinds of packages to use?

Turn and talk to a partner. Take turns answering these questions. Share your responses to the second question.

Think about your favorite kind of candy.

DOING MATHEMATICS “FINDING THE BEST BOX”

Read the complete instructions in Handout 2-16: Thinking Geometrically. Our company, Out of This World

Candies, plans to sell our Starburst candies in a new package containing 24 individually wrapped Starbursts. Your challenge is to find the dimensions of the least expensive box that can hold exactly 24 Starbursts.

Write a report including the dimensions for all the possible boxes we can use to package 24 Starbursts, the dimensions of the least expensive box, and explain how you know your answers are correct.

You and your teammates represent the Best Solutions Consulting Company.

Out of This World Candies wants you to solve the following problem:

2-16

Each wrapped Starburst has a square shape that measures 2 cm on a side and 1 cm high.

2 cm

2 cm 1 cm

The following video clip shows an 8th grade teacher introducing “Finding the Best Box” to her students.

A Classroom Example of Doing MathematicsFinding the Best Box

The School• Urban school• 90% English language

learners

The Class• 8th grade class, • Most students are English language learners who have “transitioned out”

of a sheltered ESL class during the past year or two.

The Teacher• Is in her 4th year of teaching• This is the first time she has used this activity which

she developed to challenge her 8th graders to apply their knowledge of geometry

As you watch this video clip think about:

• How did this teacher create a context to engage her students?

• What do you wonder about?

Use HANDOUT 2-17 to record your observations.

A Classroom Example of Doing MathematicsFinding the Best Box (continued)

View the Video Now

2-17

After viewing the video and writing your observations and questions Turn and Talk with a partner. Discuss what you have both written on HANDOUT 2-17.

SHARE

A Classroom Example of Doing MathematicsFinding the Best Box (continued)

DOING MATHEMATICS “FINDING THE BEST BOX”

Read the complete instructions in Handout 2-16: Thinking Geometrically. Our company, Out of This World

Candies, plans to sell our Starburst candies in a new package containing 24 individually wrapped Starbursts. Your challenge is to find the dimensions of the least expensive box that can hold exactly 24 Starbursts.

Write a report including the dimensions for all the possible boxes we can use to package 24 Starbursts, the dimensions of the least expensive box, and explain how you know your answers are correct.

You and your teammates represent the Best Solutions Consulting Company.

Out of This World Candies wants you to solve the following problem:

2-16

Each wrapped Starburst has a square shape that measures 2 cm on a side and 1 cm high.

2 cm

2 cm 1 cm

“FINDING THE BEST BOX” (continued)

2-16

Your task is to write a report including:

1. The dimensions in centimeters for all the possible boxes we can use to package exactly 24 Starbursts.

2. The dimensions of the least expensive box – the cheapest one for us to make.

3. An explanation stating how you know your answers to questions 1 and 2 are complete and accurate

4. A suggestion to us about which box you think would be our best choice. We want to know why you think a particular box is the best choice over all the others.

1st step: Unpacking the Problem

Take a few minutes to re-read the problem on HANDOUT 2-16. Think individually about each topic listed below.

What SPECIFIC INFORMATION is given in the problem?

What PRIOR KNOWLEDGE can we use to solve the problem?

What do we need to FIND OUT that will help solve the problem?

2-16, 2-19, 2-20, 2-21

1st step: Unpacking the Problem (continued)

Use HANDOUT 2-19. Individually write your answer to the question: What SPECIFIC INFORMATION is given in the problem?

Then take turns using Speak, Listen, Question and Respond to talk with a partner about the specific information given.

Speak

Listen

QuestionRespond

The problem tells us that there are 24 Starbursts in a box.

Hmm ... She talked about how

many Starbursts

are in a box.

Why is that important?

Because that will tell us

how big the boxes have to

be.

SHARE with the large group.

2-16, 2-19


Use HANDOUT 2-20. Individually write your answer to the question: What PRIOR KNOWLEDGE can we use to solve the problem?

Then take turns using Speak, Listen, Question and Respond to talk with a partner about the prior knowledge in the problem.


2-16, 2-20


Use HANDOUT 2-21. Individually write your answer to the question: What do we need to FIND OUT that will help us solve the problem?

Then take turns using Speak, Listen, Question and Respond to talk with a partner about what we need to find out to solve the problem.


2-16, 2-21

2nd step: Partial Solutions

2-22

CHOOSE a STRATEGY to solve a problem

Draw a Picture

Create a Model

Make a list, table or

chart

Look for a PatternUse Easier Numbers

1, 2, 3

10, 20, 30

Write an equation

a2 + b2 = c2

Work backwards

2nd step: Partial Solutions

Work by with your partner to find the dimensions in centimeters of one box that can hold exactly 24 Starbursts. First Choose a Strategy (HANDOUT 2-22) you will use to find the dimensions.

When you have found the dimensions of one box draw a large diagram of the box on a sheet of blank paper, showing its length, width and height in centimeters.

Record your results HANDOUT 2-23 and post them on a chart where the entire group can see them.

DIMENSIONS DIAGRAMS

8 cm x 6 cm x 2 cm

8 cm6 cm

2 cm

2-22, 2-23

2nd step: Partial Solutions (continued)

Look at all the dimensions of the boxes posted so far.

Think and write (HANDOUT 2-24)

Do you think we have found all the possible boxes? Explain: give reasons for your answer.

How many more do you think there might be: hundreds of possibilities or just a few? Explain: give reasons for your answer.

2-24

2nd step: Partial Solutions (continued)

Listeners Choose a scaffolding question from the bottom of the handout.

Next SpeakerTry to answer each person’s question. Take turns speaking and asking questions until the group has reached consensus.

GET INTO TEAMS OF FOUR.

First speakerShare what you have written on HANDOUT 2-24 using

“I think there are …

no a few many

… more possible boxes because …”

2-23, 2-24

Using Scaffolding Questions

I think there are many more possible boxes because we’ve got only three different ones so

far.

Could we make a stack even taller?

I can visualize a tall stack of 24 Starbursts. The stack will fill a box that’s 2 cm long, 2 cm wide and 24 cm high

Can someone give me an example?

#1#2

#3#4

2-23, 2-24

When you have reached consensus on this one question, share with the larger group.

3nd step: Completing the Solution

FINDING ALL THE BOXES

TEAM:Work with your teammates to make a plan for finding all

the possible boxes that can hold exactly 24 Starbursts.

As you make your plan, think about the following questions:

How will we organize our work to keep track of all the boxes we have found?

How will we know for sure that we have found all the possible boxes?

SHARE YOUR PLAN

2-23, 2-24

3nd step: Completing the Solution (continued)


TEAM• Use your plan to find as many different boxes as you can.

• Every time you find a new box post a diagram of it and its dimensions on the chart for the whole group.

• Continue to use the scaffolding questions on HANDOUT 2-24 informally as you work towards finding all the boxes.

DIMENSIONS DIAGRAMS

8 cm x 6 cm x 2 cm

8 cm6 cm

2 cm

2-23, 2-24

3rd step: Completing the Solution (continued)

HANDOUT 2-25 shows some of the data gathered by teachers in an earlier workshop.

TEAM

What pattern or patterns do you notice in the data?

Can these patterns help you determine whether you have found all the possible boxes that can hold exactly 24 Starbursts?

2-25


TEAMTalk with your team. Review the list of boxes and dimensions posted so far.

Are you all convinced that you have found all the possible boxes that will hold exactly 24 Starbursts? Can you make a convincing argument about this within your team?

Try to reach consensus on this point.


Finding all the boxes

SHAREDoes the entire workshop group agree that all possible boxes have been found?


Finding all the boxes


WRITE: Individually answer the question at the top of HANDOUT 2-26.

FINDING THE LEAST EXPENSIVE BOX

How can we determine which box is least expensive to produce?

2-26


TEAMFirst speaker.

Use this pattern language to share what you have written on HANDOUT 2-26:

To find the least expensive box I suggest that we _________________ because ____________________.

Listeners Choose a scaffolding question from HANDOUT 2-26

Next SpeakerAnswer each person’s question. Take turns speaking and asking questions until the group has reached consensus on a plan.

SHARE


2-26


TEAMWithin your team, divide responsibility for completing the calculations needed to find the least expensive box.

• How will you make sure the calculations are complete?• How will you check for accuracy?• You may want to use a calculator.

SHARE YOUR RESULTS WITH THE ENTIRE WORKSHOP GROUP.

Ask clarifying questions of the other teams to make sure the work is complete and accurate and to make sure everyone understands what each group has done.

2-26



THINK AND WRITE:Individually, use HANDOUT 2-27a and 2-27b to plan your group’s report to Out of This World Candies. Re-read HANDOUT 2-16 to remember what the company asked for.

TEAM Take turns sharing items from your lists and deciding whether to include each item in the report. Together, write a list of what you think the report should contain.

SHARE

Planning a Final Report

2-27

Reflecting on Doing Mathematics – Math Content

Reflecting on Mathematical ContentThink about the following questions, first individually

and then with a partner.

THINK and WRITE:• Did anything about the mathematics of this problem surprise

you?• Did you recognize any mathematical misconceptions in your own

and your colleagues work? What mathematical questions do you have after solving this

problem? What did you learn about the way you and others approach

mathematical problems?

PAIR:Share your answers with your partner.

SHARE something significant that your partner said.

2-28

Think

Write

Pair

Share

Reflecting on Doing Mathematics -- Process

Reflecting on the Process of Doing Mathematics

Think about the following questions, first individually and then with a partner.

THINK and WRITE: Did anything about this process surprise you? How did group members help every member

understand and solve the problem? How is “doing mathematics” similar and different

from the ways these kinds of problems are normally taught?

PAIR:Share your answers with your partner.

SHARE something significant that your partner said.

2-28

Think

Write

Pair

Share

Reflecting on “Doing Mathematics” and English language learners

1. How does this lesson ensure that English Language Learners are engaged in high-level Mathematics:

solving challenging problems, using mathematical reasoning, and explaining their thinking?

2. How does this lesson insure that all English Language Learners are engaged at all times throughout the learning process?

3. How does this lesson support English language learners learning mathematics in the middle grades

without simplifying the problem, telling them what to do, or telling them the answer.

Three Essential Questions about Planning Lessons for English

Language Learners

2-29

Essential Questions for Lesson Planning



2-29

Think

Write

Pair

Share

Write: Individually write your responses.

Pair: Discuss your responses with your partner.

Share with the whole group.

CHOOSE A NEW PARTNER

Think: Take about a minute to think about the first question as it applies to the activity we just completed.


(continued)

2. How does this lesson insure that all English Language Learners are engaged at all times throughout the learning process?

2-29

Think

Write

Pair

Share




Think: Take about a minute to think about the second question as it applies to the activity we just completed.


(continued)



2-29

Think

Write

Pair

Share




Think: Take about a minute to think about the third question as it applies to the activity we just completed.

Keeping Classroom Activities at a High Level

of Cognitive Demand

Low-level math

High-level Math

Mathematical tasks can be categorized by Levels of Cognitive Demand

• Low level tasks involve Memorizing Facts and Following Procedures

• Intermediate level tasks involve Using Procedures and Concepts

• The highest level tasks involve Doing Mathematics, that is, solving novel and challenging problems

2-30

Identifying Levels of Cognitive Demand

As teachers we need to be able to identify which level a particular task is at so that we can provide our English language learners and all students with the high-level mathematical tasks.

If we determine that a task in our curriculum is at a low-level, we can adapt it to increase its cognitive demand and enhance the learning of our students.

Identifying Levels of Cognitive Demand2-30, 2-31

THINK and WRITE:

Consider the three tasks on HANDOUT 2-31. Tasks A, B and C all focus on ratio and proportion.

•Read all three tasks, by yourself.

•Use the rubric on HANDOUT 2-30 to decide the level of each task

•Write in the box next to each task:

• The level of cognitive demand

• Your reasons for placing it at that level.


TEAM:

Form groups of three.

Use the Final Word protocol to discuss this with your team mates. Each of you will be the first speaker for one of the three tasks.


TEAM:

The first speaker says which level they chose for Task A and explains why. The other speakers respond in order to agree, disagree and comment. The first speaker has the Final Word for Task A.

Repeat the process starting with a different first speaker for Tasks B and C.

I think task A is at level …

Agree or disagree and comment

Agree or disagree and comment

Final Word

#1 #2 #3 #1

SHARE

Modifying Levels of Cognitive Demand2-32

The cognitive demand of a task can be changed. It can be raised to make a task more challenging. It can be lowered to make a task simpler and more routine.

Consider Task B ( a Level 1 task):

3/5 of the students in Ms. Jones’ class of 30 students are boys. How many of the students are boys and how many are girls?

TEAM

Brainstorm at least three different ways that Task B can be modified to make it more challenging. Write your responses on HANDOUT 2-32.

SHARE

Modifying Levels of Cognitive Demand

2-32

Consider Task A ( a Level 2 task):

Students at Grayson Middle School are ordering school T-shirts. They come in two colors. A survey of 25 students shows that 10 students preferred blue T-shirts and 15 students preferred red. Explain how you can estimate how many red and blue T-shirts to order, if a total of 180 students order shirts. Use diagrams, tables and mathematical expressions to make your explanation easier to understand.

TEAM

Brainstorm at least three different ways that Task A can be modified to make it more challenging. Write your responses on HANDOUT 2-32.

SHARE

Modifying Levels of Cognitive Demand2-32

Consider Task C ( a Level 3 task):

The student council has budgeted $300 to buy drinks for the graduation dance. Your job is to order drinks that most students will like. A survey of 40 students shows that 15 prefer cola, 5 prefer ginger ale and the rest prefer lemonade. Sodas cost $2.00 for a 2-liter bottle and lemonade costs $3.00 for a 2-liter container.

To have enough drinks for everyone, you want to spend as much of the $300 as possible without going over budget. Decide how much of each drink to order and write a report to the student council justifying your decision.

TEAM

Task C is written at a high-level of cognitive demand. But it is possible to simplify a task and make it less challenging. Often this happens inadvertently as teachers try to help their students.

Brainstorm at least three different ways that Task C can be simplified to make it less challenging. Write your responses on HANDOUT 2-32.

SHARE

How Tasks Change When They are Implemented in the Classroom

We have seen that teachers can intentionally modify curriculum tasks to make them more challenging.

However, research has shown that teachers and students often lower the cognitive demand level of a mathematical task, often without being aware that they are doing so.

2-33

How Tasks Change When They are Implemented in the

Classroom

Student Learning

Tasks as they

appear in curriculu

m materials

Tasks as set up by teachers

Tasks as enacted

by teachers

and Students

Consider this schematic diagram*:

2-33

* Adapted from: Implementing Standards-Based Mathematics Instruction: A Casebook for

Professional Development. Teachers College Press, 2000, by M. K. Stein, M. S. Smith, M.A. Henningson and E. A. Silver.

How Tasks Change When They are Implemented in the Classroom

Tasks as they

appear in curriculu

m materials

A task – such as Finding the Best Box – may be set up at the Doing Mathematics level of cognitive demand:

• Students are asked to solve a novel problem using knowledge of mathematical concepts and problem solving strategies.

• They are asked to develop their own approach to the problem and explain their thinking.

2-33

How Tasks Change in the Classroom

Tasks as they

appear in curriculu

m materials

However, teachers may decide to simplify the task for their students.

• They may tell students what to do first, second, third.

• They may tell students, for instance, that they have have to find the box with the smallest surface area.

• They may provide formulas for students to use, give them a table to fill in, etc.

• They may eliminate the requirement for students to explain their work both orally and in writing.


2-33


Teachers commonly simplify challenging mathematical tasks for English language learners!

Why do you think this is true?

Turn and Talk with a partner about this.

2-33

SHARE


Tasks as they

appear in curriculu

m materials

Furthermore, when students ask for help teachers may simplify the task even more by giving hints or asking leading questions.

• They may tell students that they have (or have not) found all the possible boxes rather than making them reason it out for themselves.

• They may tell students to first find the volume of 24 Starbursts, and then use the factors of that number to find dimensions of other boxes..

• They may give “hints” or “remind” students to use formulas they already know for volume and surface area.


Tasks as enacted

by teachers

and Students

2-33


Student Learning

Tasks as they

appear in curriculu

m materials


Tasks as enacted

by teachers

and Students

Most teachers simplify problems for their students. Some do it more often than others.

Many teachers simplify problems more often for English language learners than for native English speakers.

Why do you think this is so?

2-33


Student Learning

Tasks as they

appear in curriculu

m materials


Tasks as enacted

by teachers

and Students

Think of a time when you simplified a problem for your students by telling students how to approach a problem or giving them hints that led them to the procedure you wanted them to follow, or to the answers?

Think Write Pair Share

2-33


The types of changes we’ve been discussing can have the net effect of reducing the cognitive demand of a task that was designed to be at the level of Doing Mathematics, to the lowest level, following step-by-step procedures and plugging numbers into formulas.

Teachers who make such changes mean well. They want to help students “succeed” in solving a particular problem with less frustration, but they deprive them of opportunities to learn high-level mathematics.

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To summarize:


Teachers of English language learners must make sure their students have access to high level mathematics in the middle grades. To do this they must guard against the cognitive demands of mathematical tasks the set for their students.

To Summarize (continued):

Supporting High-Level Learning of Mathematics for English Language Learners

If we don’t help our students by simplifying a problem, showing them which steps to take, or giving them hints, how can we help them avoid frustration when they are stuck?

Supporting High-Level Learning of Mathematics for English Language Learners

This is the basic definition of scaffolding.

Our challenge as teachers of English language learners is to scaffold their learning without simplifying their tasks.

We have to find ways to support their learning without

•lowering the cognitive demand,

•simplifying the problem,

•telling them what to do, or

•telling them the answer.

Scaffolding High-Level Learning of Mathematics for English Language Learners

Work with a partner. Make a list of at least 3 ways to scaffold “Finding the Best Box” for English language learners.

Write your suggestions on Handout 2-34

Scaffolds for “Finding the Best Box”

Think Write Pair Share

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Scaffolding High-Level Learning of Mathematics for English Language Learners

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Read HANDOUT 2-35.

Compare the suggestions in the handout with the suggestions our group came up with.

Scaffolding Suggestions for “Finding the Best Box”

Recall the Three Essential Questions About Lesson Planning



2. How does this lesson ensure that all English Language Learners are engaged at all times throughout the learning process?



Three Essential Questions about Planning Lessons for English Language Learners

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A Few Final Thoughts

High-level mathematics in middle school is challenging, engaging, and enjoyable.

High-level mathematics instruction helps English language learners build confidence, develop problem-solving strategies, learn and practice basic skills and develop an understanding of the language, methods and purposes of mathematics.

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To keep mathematics instruction at a high level teachers must assign challenging tasks, allow students enough time to work on them, establish classroom procedures and expectations so that students work together comfortably, build on each other’s ideas and take responsibility for their own and each other’s learning.

It is challenging for teachers as well as students to maintain a high level of cognitive demand.

Scaffolding by teachers is essential in fostering the learning of high-level mathematics. This takes professional insight, preparation, and time.

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High level mathematics is within the reach of all middle grades students, of all English language learners.

We hope this challenge will become one that sustains, guides and focuses your work as a teacher.

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Action Plans: Next Steps in your own classroom

WRITE on HANDOUT 2-37 one best practice from this workshop for supporting high-level mathematical learning with English language learners that you plan to try out with your own students. PAIR with a partner to discuss the practice and how you plan to implement it.

Plan how you will collect evidence (student work, teacher lesson plans, video, audio, etc.) of how well it worked. Use HANDOUT 2-38 “Changes I have made in my practice” to document what happens.

2-37, 2-38

Workshop 1

Teaching High-Level Mathematics to English Language Learners in the Middle Grades


Thank you for attending the workshop

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Please help us out by completing the Workshop Evaluation, Handout 2-39 before you leave.

Project Development Team

‣ Turning Points, Center for Collaborative Education Dr. Sara Freedman, Project Developer Dr. Dan Lynn Watt, Math Consultant

‣ Teachers, Burncoat School, Worcester Massachusetts Judith Murphy

Tracy Olearczyk

‣ Teachers, Calcutt School, Central Falls Rhode Island Cathy Carvalho Jennifer Martin Jillian O’Keefe


mathematics for ell students workshop 2 presentation

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