math for ell students workshop 2 handouts
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Teaching
High-Level Mathematics
to English Language
Learners
in the Middle Grades
Participant’s
Packet
Worksho 2
Copyright 2009 Center for Collaborative Education/Turning Points
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Prepared for
Turning Points
by
Dr. Sara Freedman, Project Director
Dr. Dan Lynn Watt, Math Consultant
Copyright 2009 Center for Collaborative Education/Turning Points
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Teaching High‐Level Mathematics to English Language Learners in the
Middle Gradeswasdevelopedby Turning Points, a project atthe
CenterforCollaborativeEducationinBoston,MA.Thistoolispartof
the Mathematics Improvement Toolkit , a project of the National
Forum toAccelerateMiddleGradesReform, andwas supportedbythe U.S. Department of Education’s Comprehensive School Reform
Initiative,grant#S332B060005.Opinionsexpressedarethoseofthe
authorsandarenotnecessarilythoseoftheDepartment.
Copyright 2009 Center for Collaborative Education/Turning Points
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Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
Warm-up/Checking In Big Ideas of Workshop 2 – Mathematical
Tasks and Cognitive Demand Comparing the Cognitive Demands of
Two Tasks: Linguistic, mathematical andcultural
Doing Mathematicso Classroom Video #1o Doing Mathematics: Finding the Best
Boxo Reflecting on Doing Mathematics
Keeping Classroom Activities at a High
Level of Cognitive Demand Action Plans: Next Steps in Your Own
Classroom Closing Circle
Handout 2-1
Agenda for Workshop 2
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Handout 2-2Overview of Workshop Series
In workshop 1, you learned:
• who are the English language learners in schools
today
• 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 mathematicareasoning 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 high-level math activities that
engage English Language Learners in the middle
grades.
Turning Points Tool Teaching
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Mathematics
o English Language
Learners
n the Middle Grades
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Handout 2-3Best Practices Modeled in
Workshop 1
Turning Points Tool Teaching
High-Level
Mathematics
o English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poi
• Using prior knowledge
• Providing time and support for English
language learners to discuss, process and explain
their mathematical reasoning
• Creating predictable routines and signals
• Checking for understanding throughout the
lesson from all students
• Using discussion protocols
• Using structured cooperative learning
• Think/write/pair/share
• Assigning numbers for round robin discussion
• Using visual cues
• Providing and practicing sentence frames
•
Speak/respond/follow-up question/respond• Organizing template/graphic organizer
• Using numbered heads as an assessment and
sharing strategy
• Final word
• One new learning at a time
• Providing concrete representations
• Creating models
• Role playing
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Handout 2-4
1. Think about one best practice from the previous workshop that
you have used to change the way you teach English language
learners - a practice that you have added to your personaltoolkit.
(You may use the list of best practices reproduced in Handout 2-
3.)
2. Write it down, explaining how it has changed your teaching and how it hashelped the English language learners in your classroom to learn high-levelmathematics.
(You may use Handouts 1-24 and 1-25 introduced at the last
workshop to help you refresh your memory if you brought themwith you. )
Turning Points Tool Teaching
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Mathematics
to English Language
Learners
in the Middle Grades
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Handout 2-6
Notes from the Share Out Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
Why did you choose this particular strategy to try out?
What did you learn by sharing with a partner?
************************************************************
(Your own questions and further notes)
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Turning Points Tool
to Teaching
High-Level
Mathematics
to English Language
Learnersn the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
The Big Ideasof Workshop 2
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Handout 2-7Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
The cognitive demand of a task is the sum total of what
a student needs to know, understand and be able todo 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.
“… 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.”*
Key Finding of the Quasar Project:
*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.
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Handout 2-8
Final Word Protocol
© Copyright 2009 Center for Collaborative Education/Turning Po
Pair with another person. Decide who will speak first.
1. Each of you choose a brief excerpt from the quote: a
phrase or short passage to discuss. Take some time to
think about why it is important to you. Take some notes if
you wish.
2. Person#1 reads their phrase and explains what it means tothem 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.
3. Person #2 retells what they heard person #1 say and adds
their own comment.
4. Person #1 has the last comment, the Final Word, taking
into account all that has been said.
5. Switch roles and repeat the protocol starting with
’
Adaptations for English language learners
Give students sentence starters or pattern language to use indiscussing their excerpt.
I think _____ is important because...
I heard you say that.... One connection I can make is...
(See appendix for additional sentence starters.)
BEST
PRACTICE:FINAL WORD
efficient way to
discuss a
reading
providescontrolled
choice makes sure all
voices are
heard ensures in-
depth
discussion ofissues
important to
group
provides entry
point intodiscussion forthose who
have not been
able to read or
absorb entire
document
Englishlan ua e
#1 #2 #1
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© Copyright 2009 Center for Collaborative Education/Turning Poi
Handout 2-9
Compare two tasks
Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
2 cm
2 cm
1 cm
Task 1. Find the surface area and volume of arectangular prism that measures 2” x 4” x 24”.
Task 2. Out of This World Candies plans to sellStarburst candies in a new package containing 24individually wrapped Starbursts. Your challenge is to
find the dimensions of the least expensive box thatcan hold exactly 24 Starbursts. Each wrappedStarburst has a square shape that measures 2 cm on aside and 1 cm high.
Write a report including the dimensions for all thepossible boxes we can use to package 24 Starbursts,the dimensions of the least expensive box, andexplain how you know your answers are correct.
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Best Practice:• Ensures both parties
are involvedthroughout theconversation. No one
can dominate.• RETELL rovides a
Handout 2-10 Linguistic Demands
Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
Task 1. Find the surface areaand volume of a rectangularprism that measures 2” x 4” x24”.
Task 2. Out of This World Candies plans to sellStarburst candies in a new package containing 24individually wrapped Starbursts. Your challenge is find the dimensions of the least expensive box thacan hold exactly 24 Starbursts. Each wrapped
Starburst has a square shape that measures 2 cm side and 1 cm high.Write a report including the dimensions for all thepossible boxes we can use to package 24 Starbursthe dimensions of the least expensive box, andexplain how you know your answers are correct.
Linguistic demands
Task 1 Task 2
© Copyright 2009 Center for Collaborative Education/Turning Po
.
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Handout 2-11 MathematicalTurning Points Tool to Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
Task 1. Find the surface areaand volume of a rectangularprism that measures 2” x 4” x
24”.
Task 2. Out of This World Candies plans to sellStarburst candies in a new package containing 24individually wrapped Starbursts. Your challenge is tfind the dimensions of the least expensive box that
can hold exactly 24 Starbursts. Each wrappedStarburst has a square shape that measures 2 cm oside and 1 cm high.Write a report including the dimensions for all the
possible boxes we can use to package 24 Starburststhe dimensions of the least expensive box, and expl
how you know your answers are correct.
Mathematical demands
Task 1 Task 2
© Copyright 2009 Center for Collaborative Education/Turning Po
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Handout 2-12 Cultural Demands Turning Points Tool
Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
Task 1. Find the surface areaand volume of a rectangularprism that measures 2” x 4” x24”.
Task 2. Out of This World Candies plans to sellStarburst candies in a new package containing 24individually wrapped Starbursts. Your challenge isfind the dimensions of the least expensive box thacan hold exactly 24 Starbursts. Each wrapped
Starburst has a square shape that measures 2 cm side and 1 cm high.Write a report including the dimensions for all the possible boxes we can use to package 24 Starburst
the dimensions of the least expensive box, and exphow you know your answers are correct.
Cultural demands
Task 1 Task 2
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Handout 2-13
JIG SAW ACTIVITY PROTOCOL – TASK 1
Handout 2-13
JIG SAW ACTIVITY PROTOCOL – TASK 1
1. Divide into three expert groups; one each forlinguistic, mathematical and cultural demands. *
2. Each person will share his/her answers, one at atime, with the “expert” group.
Come to consensus on the best responses
• consider all responses respectfully,
• ask each other questions, and
• decide which ones are the best.
Turning Points Tool Teaching
High-Level
Mathematics
o English Language
Learners
n the Middle Grades
All members of this group must make sure thateveryone in the group is an expert on the particulardemand being considered by the end of this part ofthe activity.
3. Responses of my expert group on ____________ demands forTask 1. (Copy the information on your groupʼs chart.)
Task 1. Find thesurface area andvolume of arectangular prismthat measures 2” x4” x 24”.
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Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
* Make sure that English language learners have sentenceframes that they can use to answer the questions and havepracticed the key vocabulary words they will need to use in theirgroups.
Handout 2-14
JIG SAW ACTIVITY PROTOCOL – TASK 2
1. Divide into three expert groups; one each forlinguistic, mathematical and cultural demands. *
2. Each person will share his/her answers, one at atime, with the “expert” group.
Come to consensus on the best responses
• consider all responses respectfully,
• ask each other questions, and
• decide which ones are the best.
urning Points Tool Teaching
High-Level
Mathematics
o English Language
earners
n the Middle Grades
All members of this group must make sure thateveryone in the group is an expert on the particulardemand being considered by the end of this part ofthe activity.
3. Responses of my expert group on ____________ demands fortask 2. (Copy the information on your groupʼs chart.)
Task 2. Out of This
World Candies plans to sellStarburst candies ina new packagecontaining 24individually wrappedStarbursts. Yourchallenge is to findthe dimensions of
the least expensivebox that can holdexactly 24Starbursts. Eachwrapped Starbursthas a square shapethat measures 2 cmon a side and 1 cmhigh.
Write a report
including thedimensions for allthe possible boxeswe can use topackage 24Starbursts, thedimensions of theleast expensive box,and explain how youknow your answersare correct.
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Handout 2-15
Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
Task 1. Find the surface areaand volume of a rectangularprism that measures 2” x 4” x24”.
Task 2. Out of This World Candies plans to sellStarburst candies in a new package containing 24individually wrapped Starbursts. Your challenge is find the dimensions of the least expensive box tha
can hold exactly 24 Starbursts. Each wrappedStarburst has a square shape, that measures 2 cm a side and 1 cm high.Write a report including the dimensions for all the possible boxes we can use to package 24 Starburst
the dimensions of the least expensive box, and exphow you know your answers are correct.
© Copyright 2009 Center for Collaborative Education/Turning Po
Consider the key finding of the Quasar Project:
“… students who performed best … were in classrooms in which task
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 stude
thinking, reasoning, problem solving and communication.”
If you want English language learners to engage in high-level mathematics abecome successful mathematical thinkers, reasoners, problem solvers andcommunicators, would you use Task 1 or Task 2? Explain why.
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Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
Doing Mathematics!with
English Language Learners
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Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learnersn the Middle Grades
Handout 2-16Thinking Geometrically: Finding the best box
This is a classic geometric problem that can help studentsdevelop and consolidate their understandings of basic
concepts of two and three-dimensional geometry, and of hogeometry connects to basic arithmetic.
The Problem Statement:
To: Best Solutions Consulting Company
From: Out of This World CandiesRe: Problem to be solved
Our company, Out of This World Candies , plans to sell our Starburst candies in anew package containing 24 individually wrapped Starbursts. Your challenge is to findthe 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 cmhigh.
In your report we want you to tell us:
1. The dimensions in centimeters of all the possible boxes we cause to package exactly 24 Starbursts.
2. The dimensions of the least expensive box for us to produce.3. An explanation of your answers to parts 1 and 2.4. A suggestion to us about which one, of all the boxes, you thin
would be our best choice. We want to know why you think aparticular box is the best choice over all the others.
* Out of This World Candies is an imaginary company. Starburst Fruit Chews are a real product
manufactured by Mars, Inc.
© Copyright 2009 Center for Collaborative Education/Turning Poi
2 cm
2 cm 1 cm
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Handout 2-17Classroom Video: Introducing the Starburst Problem to EnglishLanguage Learners
Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poi
How does the teacherengage her students?
What do I wonder?
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Handout 2-18Speak, Listen, Question and Respond
Turning Points Tool Teaching
High-Level
Mathematics
to English LanguageLearners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
Read HANDOUT 2-16 and use HANDOUTS 2-19, 2-20, and 2-21
One at a time, individually, complete each of the three handouts:
2-19: What SPECIFIC INFORMATION is given in the problem
2-20: What PRIOR KNOWLEDGE can we use to solve the problem
2-21: What do we need to FIND OUT that will help us solve theproblem?
Divide into pairs. Use SPEAK, LISTEN, QUESTION and RESPONDwith your partner to share and discuss your responses. When it is youturn to ask questions, you may use the scaffolding questions onHANDOUTS 2-19, 2-20 and 2-21 to question your partner.
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Handout 2-19
Turning Points Tool Teaching
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Mathematics
to English Language
Learnersn the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
Sacffolding questions to ask your partner:
Where does it say that in the problem?Why is that important?How does knowing that help us solve the problem?Please explain or show what [a word or phrase] means.
Statements to make to your partner:
The problem tells us that.....It also tells us that....Another fact included in the problem is that....Finally, the problem tells us that...
What SPECIFIC INFORMATION is given in theroblem?
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Handout 2-20Turning Points Tool Teaching
High-Level
Mathematics
o English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
Statements to make to your partner:
I already know that ...I know how to find ...Finally, I know that ...I know this problem is the like ... because...Therefore, I know that I can use ... to help solve the
problem.
What PRIOR KNOWLEDGE can we use to solve theproblem? (What do we already know that can help us?
Scaffolding questions to ask your partner:
How do you know that?How does that relate to the mathematics of this problem?How will that help us solve the problem?Please explain or show what [a word or phrase] means
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Handout 2-21
Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
What do we need to FIND OUT that will help solve theproblem?
Statements to make to your partner:
We need to find out ...We also need to know ... because …Finally, we need to answer the question ... in order tofinally solve the problem.
Scaffolding questions to ask your partner:
How will we know if we have solved the problem?Can you explain why we need to know that?How will knowing that help us find the solution?How will we be able to find that out?Please explain what [a word or phrase] means.
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Handout 2-22Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poi
Draw a picture Make a list, table or cha
Create a model
Use easier numbers Look for a pattern
1, 2, 3
10, 20, 30
Write an equation Work backwards
a2 + b2 = c2
CHOOSE a STRATEGY to solve a problem
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Handout 2-23Keeping Track of InformationUse more co ies of this a e if necessar
Turning Points Tool
Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
DIMENSIONS DIAGRAMS
8 cm x 6 cm x 2 cm
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Handout 2-24Evaluating Partial Solutions
Turning Points Tool Teaching
High-Level
Mathematics
o English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Look at all the boxes and dimensions posted byever one so far and answer the followin
Statements to make to your partner:no
I think there are … a few more boxes possible because ...many
Do you think we have found all the possible boxes?
How many more do you think there might be: hundreds of
possibilities or just a few?
Explain: give reasons for your answers.
Scaffolding questions to ask your partner:
How many of the boxes that people already found are the same?how many are different?
Do you think there could be any more boxes longer (shorter, highthan any of these?
What’s the longest (tallest, shortest ) box we could make?
Can you describe any patterns among the boxes that are listed?
Do you think there are any numbers that cannot be used as one othe dimensions of a box that will hold exactly 24 Starbursts. Whatare some of those numbers? How do you know?
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Handout 2-25
Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learnersn the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Examine this data gathered by teachers in an earlierworkshop.
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?
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rning Points Tool eaching
gh-Level
athematics
English Language
arners
the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
Now that you have found all possible boxes that can hold
exactly 24 Starbursts, how can you determine the leastexpensive box?
Handout 2-26Finding the Least Expensive Box
Scaffolding Questions
What calculations do you need to make?
Do you need to make a calculation for every possible box, or just some of them? Explain.
Could there be more than one box that is least expensive?
What argument can you make to convince someone that youhave found the least expensive box?
Pattern Language:
To find the least expensive box I suggest that we ____ because _____.
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Handout 2-27aReporting to Out of This World Candies
Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learnersin the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
To: Best Solutions Consulting CompanyFrom: Out of This World Candies
Re: Problem to be solved
Our company, Out of This World Candies , plans to sell our Starburst candies in anew package containing 24 individually wrapped Starbursts. Your challenge is to find thdimensions 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 cmhigh.
In your report we want you to tell us:
1. The dimensions in centimeters of all the possible boxes we can useto package exactly 24 Starbursts.
2. The dimensions of the least expensive box for us to produce.3. An explanation of your answers to parts 1 and 2.4. A suggestion to us about which one, of all the boxes, you think
would be our best choice. We want to know why you think aparticular box is the best choice out of all the others.
Here is an excerpt from the Memo from Out of This World Candiesto Best Solutions Consulting Company
Make a list of items to include in your report to Out of This World
Candies. Pay particular attention to items 3 and 4. For item 3, be sure you make a concise and convincing case that you
have found the information requested. For Item 4, you may want to recommend that Out of This World
Candies use a box that is NOT the least expensive box. If you makesuch a recommendation be sure to think about:
o How would it look to shoppers?o What would make it appealingo What kind of visual information should be displayed on the
box?o How would the box look on a shelf with other candies?
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Handout 2-27bReporting to Out of This World Candies
Turning Points Tool Teaching
High-Level
Mathematics
to English LanguageLearners
in the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Use this page to write down the items you plan to includein your report.
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Handout 2-28Debrief: Reflecting on “doing mathematics” in thiworkshop Turning Points Tool
Teaching
High-Level
Mathematics
o English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
Think about the following questions within your team and with the entire worksgroup. Answer these questions as workshop participants, not as English langulearners in the middle grades.
1. Mathematical content: Did anything about the mathematics of this problem surprise you
Did you recognize any mathematical misconceptions in your owand your colleague’s work? If so, how were they addressed?
What mathematical questions do you have after solving thisproblem?
What did you learn about the way you and others approachmathematical problems?
2. The Process of Doing Mathematics: Did anything about this process surprise you?
How did group members help every member understand and sothe problem?
How is “doing mathematics” similar and different from the waysthese kinds of mathematical concepts are normally taught?
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Handout 2-29Debrief: Reflecting on “doing mathematics” withEnglish language learners in the classroom
Turning Points Tool Teaching
High-Level
Mathematics
o English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Poin
Three Essential Questions for Lesson Planning
1. How does this lesson ensure that English languagelearners are engaged in high-level mathematics by
solving challenging problems, using mathematical reasoning, and explaining their thinking?
2. How does this lesson ensure that all English languagelearners are engaged at all times throughout thelearning process?
3. How does this lesson support English languagelearners learning mathematics in the middle grades
without simplifying the problem, telling them what to do, or telling them the answer?
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Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learners
in the Middle Grades
Keeping Classroom Activitiesat a High Level of
Cognitive Demand for
English Language Learners
© Copyright 2009 Center for Collaborative Education/Turning Po
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Turning Points Tool
Teaching
High-Level
Mathematics
to English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Handout 2-30Rubric for Levels of Cognitive
DemandLevel 1 tasks: Memorizing Facts and Following procedures
Typical activities involve: Memorizing definitions, facts, algorithms and formulas Using memorized definitions, algorithms, facts and formulas, exactly as
learned Using known algorithms to produce correct answers Following steps in a given order Evaluating an expression for a given variable Solving a one-step word problem using a known algorithm Solving problems with unambiguous correct answers Describing procedures without further explanation
Level 2 tasks: Using Procedures and Concepts
Typical activities involve Solving Multi-step problems Using and making connections among multiple representations
(diagrams, concrete models, symbols, formulas) Solving problems requiring conceptual understanding Using procedures to help illustrate or understand concepts Identifying and extending a pattern Justifying the steps in a solution Using a graph, table or figure to solve a problem
Level 3 tasks: Doing Mathematics
Typical activities involve
Solving novel problems Accessing relevant prior knowledge Analyzing multi-step mathematical tasks and recognizing their
constraints Making a plan without being given all steps or procedures ahead of time
Self-monitoring progress, evaluating and revising plans as needed Exploring to deepen understanding of mathematical concepts Using mathematical models and multiple representations to solve
problems and explain thinking Using mathematical reasoning and language, drawings and models to
explain problem-solving strategies and justify (prove) solutions Solving problems in more than one way
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Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learnersn the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Handout 2-31 Matchin Tasks and Co nitive Levels
Task DescriptionWhich Level do you think it is?Explain.
TASK A:Students at Grayson Middle School areordering school T-shirts. They come in twocolors. A survey of 25 students shows that 10students preferred blue T-shirts and 15students preferred red. Explain how you canestimate how many red and blue T-shirts toorder, if a total of 180 students order shirts.Use diagrams, tables and mathematicalexpressions to make your explanation easier tounderstand.
Task B:3/5 of the students in Ms. Jones’ class of 30students are boys. How many of the studentsare boys and how many are girls?
Task C:The student council has budgeted $300 to buydrinks for the graduation dance. Your job is toorder drinks that most students will like. Asurvey of 40 students shows that 15 prefercola, 5 prefer ginger ale and the rest preferlemonade. Sodas cost $2.00 for a 2-liter bottleand lemonade costs $3.00 for a 2-litercontainer.
To have enough drinks for everyone, you
want to spend as much of the $300 as possiblewithout going over budget. Decide how muchof each drink to order and write a report tothe student council justifying your decision.
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Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learnersn the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Handout 2-32 Modifying Cognitive Levels
Task Description
Suggestions for enhancing the
cognitive level of the task TASK A:Students at Grayson Middle School areordering school T-shirts. They come in twocolors. A survey of 25 students shows that 10students preferred blue T-shirts and 15students preferred red. Explain how you canestimate how many red and blue T-shirts toorder, if a total of 180 students order shirts.Use diagrams, tables and mathematicalexpressions to make your explanation easier to
understand.
Task B:3/5 of the students in Ms. Jones’ class of 30students are boys. How many of the studentsare boys and how many are girls?
Task C:The student council has budgeted $300 to buydrinks for the graduation dance. Your job is toorder drinks that most students will like. Asurvey of 40 students shows that 15 prefercola, 5 prefer ginger ale and the rest preferlemonade. Sodas cost $2.00 for a 2-liter bottleand lemonade costs $3.00 for a 2-litercontainer.
To have enough drinks for everyone, youwant to spend as much of the $300 as possiblewithout going over budget. Decide how muchof each drink to order and write a report tothe student council justifying your decision.
Suggestions for simplifying thistask:
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Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learnersn the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Handout 2-33How Tasks Chan e in Classrooms*
Furthermore, when students
ask for help teachers may
simplify the task even moreby giving hints or askingleading questions.
• They may tell students that they
have (or have not) found all the
possible boxes rather than makingthem reason it out for themselves.
• They may tell students to first findthe volume of 24 Starbursts, andthen use the factors of that numberto find dimensions of other boxes..
• They may give “hints” or “remind”students to use formulas they
alread know for volume and
Teachers whomake such
changes meanwell. They
want to helpstudents“succeed” in
solving aparticularproblem with
lessfrustration,
but they deprive themof
Teachers of
Englishlanguage
learners mustguard against
reducing the
cognitivedemands of
mathematicaltasks andmake sure
their students
have access tohigh- levelmathematics
in the middle
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 theyhave to find the box with the smallest surfacearea.
•They may provide formulas for students touse, give them a table to fill in, etc.
•They may eliminate the requirement forstudents to explain their work both orally
and in writing.
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
•The are asked to develo their own a roach to the roblem
* 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.
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Turning Points Tool Teaching
High-Level
Mathematics
o English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Handout 2-34Scaffolding the “Best Box” problem
for English language learners
Work with a partner to brainstorm three or more suggestions for scaffoldingFinding the Best Box to support English language learners in your ownclassrooms.
1
2
3
4
5
6
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Turning Points Tool Teaching
High-Level
Mathematics
to English Language
Learnersn the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Handout 2-35Scaffolding the “Best Box” problem
for English language learners
Suggestions for Scaffolding “Finding the Best Box”
for English language Learners
Scaffolding not only helps students solve the problem athand, but also gives them strategies they can use in any
problem-solving situation.
Compare these suggestions with the ones your workshop group came
up with: • Create a context for the problem that students will understand, even enjoy.
In our example, choosing familiar Starburst candies instead of abstract
“blocks” or “cubes” helps motivate students and keep them involved.• Provide students with sets of 24 Starbursts so they can actually construct
rectangular prisms by rearranging the candies.• Ask students to plan their work by identifying their objectives, decide what
they know and what they need to know, and make a plan for the steps theycan take to solve the problem.
• Instruct students to discuss their plans in their groups, using a structuredgroup protocol to make sure that everyone speaks and everyone listens, andcome to agreement about what they are going to do.
• Give students sentence starters (pattern language) so that they can practiceusing mathematical language when they talk in their groups or share with the
class.• Ask “scaffolding questions” that help students think, without “giving hints.”
For example: Have you ever solved a similar problem? What information do you
have to start with? What have you chosen as a first step? Have you found all the
possibilities? How do you know? Please explain your thinking. • Give students lists of scaffolding questions and encourage them to ask them
of each other.• Set up a structure to help the different groups share information so they can
all learn from the work the others have done.• Make sure that all posted work is visible to everyone, so everyone can see
the same information.
• Stop the working groups at several points to share information and learnfrom each other.
• Choose students randomly from each group to report on their group’s
decisions and discoveries. It’s important to choose randomly so that studentsknow that every student in a group is responsible for being able to explain all
the work of the group.
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Turning Points Tool Teaching
High-Level
Mathematics
o English Language
Learners
n the Middle Grades
© Copyright 2009 Center for Collaborative Education/Turning Po
Handout 2-36Some Final Thoughts
High-level mathematics in middle school is challenging,
engaging, and enjoyable.
High-level mathematics instruction helps English languagelearners build confidence, develop problem-solvingstrategies, learn and practice basic skills and develop anunderstanding of the language, methods and purposes of mathematics.
To keep mathematics instruction at a high level teachersmust assign challenging tasks, allow students enough time to
work on them, establish classroom procedures andexpectations 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 maintaina high level of cognitive demand.
Scaffolding by teachers is essential in fostering the learningof high-level mathematics. This takes professional insight,
preparation, and time.
High level mathematics is within the reach of all middlegrades 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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© Copyright 2009 Center for Collaborative Education/Turning Po
Turning Points Tool Teaching
High-Level
Mathematics
o English Language
Learners
n the Middle Grades
Handout 2-37
COMMITMENT:
Over the next few weeks I will try out thefollowing best practices for teaching high-levelmathematics to English language learners:
I will collect the following evidence in order todetermine how well it is working in improvingthe access my students have to learning high-level mathematics.
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Date: ______________________________
Best instructional practice for English language learners on whicham working:
________________________________________________________
High-level math content:
Instruction
Using mathematical languageappropriately and accurately
Using mathematical languageappropriately and accurately
Other Other
Changes I have made in mypractice
Changes I have noticed in mystudents’ depth of
understanding and engagemen
High-level math content:
Instruction:
Handout 2-38
© Copyright 2009 Center for Collaborative Education/Turning Po
Turning Points Tool Teaching
High-Level
Mathematics
o English Language
earners
n the Middle Grades
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Handout 2-39Workshop strengths (information, facilitation, pacing, mix of
activities, etc.):
Suggestions for improvement (information, facilitation, pacing,
mix of activities, etc.):
Questions or concerns I now have:
Most important ideas I will take away from the workshop:
ning Points Tool aching
gh-Level Mathematics
English Language
arners
he Middle Grades