math for ell students workshop 1 participant handouts

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

Participant’sPacket

Workshop 1

© Copyright 2009 Center for Collaborative Education/Turning Points

Prepared forTurning Points

by

Dr. Sara Freedman, Project directorDr. Dan Lynn Watt, Math consultant


Teaching High‐Level Mathema2cs to English Language Learners in the Middle Grades was developed by Turning Points, a project at theCenterforCollabora;veEduca;oninBoston,MA.This tool ispartofthe Mathema2cs Improvement Toolkit, a project of the Na;onalForum to AccelerateMiddle Grades Reform, and was supported bythe U.S. Department of Educa;on’s Comprehensive School ReformIni;a;ve,grant#S332B060005.Opinions expressedare thoseof theauthorsandarenotnecessarilythoseoftheDepartment.


Table of ContentsTurning Points ToolTeaching High-Level Mathematics to English Language Learners in the MIddle Grades


Think/Write/Pair/ShareFinal WordNumbered HeadsSpeak/Respond/Question/RespondVisual Cues

Handouts for Workshop 1

• Protocols Used in Lessons

• Introducing Pattern Language/Sentence Starters

• Giving Directions to English Language Learners

• Questions to Ask in Math Class

• Pattern Language/Sentence Starters to Use in Math Class

• Answers to Frequently Asked Questions

• Bibliography and Additional Resources

Appendix

Handout 1-1


• Challenges Facing English Language Learners: The Language of Mathematics

• Introductions/Creating Community/Using Prior Knowledge

• Classroom Video #1

• Framing Principles

• English Language Learners: Who are they today?

• Challenges: Word Problems

• Word Problems: Supports for English Language Learners

• Classroom Videos: #2, #3, and #4

• Action Plans: Next Steps in the Classroom

• Closing/Reflections/Feedback

Agenda for Workshop 1Turning Points ToolTeaching High-Level Mathematics to English Language Learners in the MIddle Grades

Handout 1-2 Overview of Workshop Series

Turning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades


In workshop 1, you will learn:• who the English language learners in schools are

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 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 high-level math activities that engage

English language learners in the middle grades

Handout 1-3ATurning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades

Think

Write✎Pair

Share

Handout 1-3B Turning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades Think about something in your own

background that can be or has already been helpful in teaching mathematics to English language learners.

WRITE your response down, using ONLY ONE SYLLABLE words. ✎

Using your prior knowledge

Handout 1-3CTurning Points Tool Teaching High-Level Mathematics to English Language Learners in the Middle Grades


Next, describe the same response using a different modality - drawing, acting out, creating a chart or map. Use as few words as possible.

drawing map graph

Handout 1-4Turning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades

What did you learn from this exercise that you will keep in mind when you teach mathematics to English language learners?

What questions did this exercise raise for you about teaching high-level mathematics to English language learners?


Do Now Problem discussed in video

Handout 1-6 Turning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades

How the teacher structures the lesson to encourage

discussion among all students

How the English language learners use language to talk

about the mathematical concepts in the lesson

Notes from Video #1


Handout 1-7 Transcript for Video #1

Teacher giving instructions to the class:

What we’re going to do is -- remember what we did yesterday? It sounded kind of funny. We were laughing at ourselves for a while. Mariah?

[inaudible response]

Those sentence starters, right? The “Say Somethings.” OK. And they sound kind of goofy when we use them to talk about the movies or stuff like that. But when we’re talking about math how we’re going to try to start using these as much as we can. So we sound like mathematicians by using a complete sentence.

So what I want you to do – on the corner of each of your desks there is a number. One person in your group has is number 1, one person has number 2, and one has number 3. Does everyone see those? OK.

If you are person 1, your job is going to be to share your answer to [problem] number 1 with the rest of your group. Yes or no, is it similar? And you’re gonna explain why. When you’re done explaining, person number 2 is going to use a “Say Something.” So, “I agree because …” or “I disagree because …” OK and then person number 3 is also going to respond. “I disagree because [sic] …” or “I disagree because ...”

Group discussion among three English language learners:Student#1: What I think about number 2, um … I think, I think it’s similar because all those numbers on the sides are the same. ‘Cause if you put these two shapes the same, like the same way as this one is, you, you could do, you could do 4 divided by 12 and that would be 3, it would be a scale factor of 3. And15 divided by 5, it’ll be 3 and the scale factor will be 3. And 3 divided by 9 will be a scale factor of 3.

#2. I agree with that, because like what you said, the scale factor is 3. 3 divided by … I mean 5 times 3 is 15, it’s all in the same spot (forming a circle with her fingers). And has the same angles in the same spot. What do you think (inaudible name)?

#3 I agree with that … because you know … if you do that … divide 4 by 3 … 4 by divided by 12 equals 3. Or 4 divided by 3 equals 12. Yeah, I agree with that. Yeah I agree.

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


Handout 1-8A Turning Points Tool Teaching High-Level Mathematics to English Language Learners in the Middle Grades

The Framing Principles

English language learners have a right to learn high-level mathematics in the middle grades.

English language learners should move forward with their conceptual math growth, while simultaneously accelerating their skills in number sense and numerical skills.

English language learners learn best in a school-wide and classroom culture of high expectations.

English language learners have a deep and broad range of prior knowledge that links directly to the critical concepts of middle grades mathematics. This prior knowledge can be used as a important resource to support the learning of all students.

Handout 1-8B Turning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades

The best instructional practices for teaching high-level mathematics to all learners are the best practices for English language learners.

English language learners need additional support in learning high-level mathematics that is uniquely designed to meet their linguistic and cultural backgrounds.

English language learners need to speak and listen to others throughout their math classes in order to master the mathematical reasoning of high-level mathematics.

English language learners must learn the language of mathematics to explain how they solved a problem and why they took the steps they took.

English language learners may use their native language as a needed support when they are first learning a concept, while at the same time developing their fluency in using the language of math in English.

The Framing Principles (continued)

Handout 1-9A Turning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades

Who are the English language learners in our schools today?English language learners are the fastest growing segment of the school population. 1 out of 10 students enrolled in public schools is an English language learner. National Clearinghouse for English Language Acquisition, 2002)

English language learners are expected to be 30% of school age population by 2015.

Nearly 1 out of 3 students enrolled in urban schools is an English language learner. National Clearinghouse for English Language Acquisition, 2002)

The percentage of English language learners enrolled in schools is increasing throughout the United States, in suburban, rural as well as urban communities. National Clearinghouse for English Language Acquisition, 2002)

Sources: U.S. Department of Education’s Survey of the States’ Limited English Proficient Students and Available Educational Programs and Services, 1991-1992 through 2000-2001 summary reports; state publications (1998-1999 data); enrollment totals from the National Center for Educational Statistics Core of Common Data, 1998-1999


Who are the English language learners in our schools today?

English language learners enroll in schools speaking many different languages. Council of Great City Schools (2003)

The great majority in the United States as a whole are native speakers of Spanish. Council of Great City Schools (2003)


Handout 1-10A Turning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades

What do we know about the experience of English language learners in our schools?

English language learners have a strong desire to receive an education. They have the highest daily attendance rate of any segment of the school population.*

English language learners have the lowest out of school suspension rates of any segment of the school population.*

However,

English language learners have the lowest standardized test scores of any segment of the school population.*

English language learners have the highest dropout rate of any segment of the school population.*•*Report of the National Literacy Panel on Language‐Minority Children and Youth (2006)

Why do you think this is so?THINK WRITE PAIR SHARE


Why do you think this is so?THINK WRITE PAIR SHARE



Learning the Language of Mathematics in the Middle Grades:

The Challenges for English Language Learners

by Sara Freedman, Ed.DThe Center for Collaborative Education

English is a language. It has its own vocabulary, idioms, and syntax. Math is a language. It has its own vocabulary, idioms, and syntax. Both of these languages are used in the middle grades math classroom. If we want English language learners to discuss and understand mathematics beyond the elementary school level, we have to teach them to think, speak, read and write in the language of mathematics as well as in conversational English.

Math class is the only place they will learn to do so and math teachers are the only teachers who can help them learn this language.

In classrooms in which students are learning high-level mathematics, the language of mathematics becomes the “native language” for all the students in that class. They learn to think, speak, read and write in that language once they enter that classroom.

Learning this new language is difficult for many students. It requires a great deal of practice and a chance to use the language in many different ways - speaking, reading, listening and writing. However, English language learners face five additional and distinct challenges in learning mathematical English that native speakers of English do not face.

1) To English language learners, the language of math may sound and sometimes look like conversational English. But the two languages are very different in several important ways that are not immediately obvious, and are often very confusing, to an English language learner.

2) English language learners in the middle grades need to learn both of these languages - conversational English and mathematical English - at the same time. They need time to process their thinking, and the culture of the class has to respect that need.

3) Many English language learners learn conversational English quickly. Their ability to use conversational English is an important resource for the math classroom, and should be encouraged. However, that fluency is often restricted to the limited vocabulary and syntax of conversational English. In fact, many English language learners’ fluency in conversational English masks their lack of fluency in the language of mathematics. Teachers may not recognize them as English language learners and may not use the kinds of supports that are critical for their continual development.

4) Language, including math language, emerges from and is heavily influenced by the culture that has created it. The language of mathematics used in the United States is different from the language of mathematics that is spoken and written in many other countries.

Handout 1-11BTurning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades


This more pro-active stance as a learner may be unsettling for many students from cultures outside the United States. They may feel that they are doing something wrong if they speak out in class, if they ask a question, if they come up with an alternative way of solving a problem. Yet English language learners need as much practice as possible speaking, listening, reading and writing in order to learn the language of math and master high-level mathematics.

5) English language learners may be unfamiliar, even uncomfortable, with the way students are expected to use language in mathematics’ classrooms in the United States, ie. that students

Learning the Language of Mathematics in the Middle Grades: The Challenges for English Language Learners

p.2

• are expected to speak often in class in small and large groups, • are expected to ask questions, • are required to give their own answers and often encouraged to come up with alternative ways of approaching a problem, • need to demonstrate their understanding of the underlying concepts of a problem, not simply list the steps of the algorithm.

Handout 1-12

What specific challenges do you think an English language learner in the middle grades might have in trying to answer the question posed by the problem?

1) What are some language difficulties in this problem for English language learners?

2) What are some math difficulties in this problem for English language learners?

3) What are some cultural features that could cause difficulty in understanding this problem for a diverse group of English language learners?

A certain construction job usually takes four workers six hours. Today, one worker called in sick, so there are only three workers. How long should it take them to do the job?



Handout 1-13


What are the lang

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fo

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What are the m

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ifficulties in this p

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for E

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What are the cultural features

that could

cause diffi

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erstanding

this pro

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for

Eng

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e learners?

A certain construction job usually takes four

workers six hours. Today, one w

orker called in sick, so there are only three w

orkers. How

long should it take them

to do the job?

1)2)3)4)5)

1)2)3)4)5)

1)2)3)4)5)

Handout 1-14AA certain construction job usually takes four workers six hours. Today, one worker called in sick, so there are only three workers. How long should it take them to do the job?



Language difficulties (some possible responses):

Conversational English versus the language of mathematicsEnglish language learners have difficulty distinguishing between words as they are used in conversational English and the same words used in the precise language of mathematics. This problem includes several examples of verbs, adjectives and adverbs, nouns and phrases that are used in a different or more precise way in the language of mathematics than in the conversational English that English language learners first acquire. (See Appendix for additional examples and resources.)

Verbstakes “takes … six hours” The verb “takes” when combined with a time, like “six hours” is used to describe the duration of an event or activity. “Take” is one of the most widely used words in English and has many meanings. The most common interpretation of “take” to imply picking up and perhaps carrying an object: “ take a card,” “ take a piece of cake,” “ take” this letter to the mailbox.”

Dictionary.com lists more than 100 definitions of “take.” Only definitions # 48 and 49 relate to the use of take in this problem: • to occupy; fill (time, space, etc.): His hobby takes most of his spare time. The machine takes a lot of room. • to use up; consume: This car takes a great deal of oil. He took ten minutes to solve the problem.

Adjectives and adverbscertain

“a certain construction job” This usage implies that we don’t need to know exactly what the job is to solve the problem. The sentence makes perfect sense without the word “certain,” which is commonly found in many word problems. So why is it there? It implies that there are many possible construction jobs, and that this is one of them. It also implies that in order to solve the problem, we don’t need to know what the job is. This may be confusing to many English language learners.

Handout 1-14BA certain construction job usually takes four workers six hours. Today, one worker called in sick, so there are only three workers. How long should it take them to do the job?




Conversational English versus the language of mathematics (continued)

UsuallyUsually – meaning “typically”, or “on average”. This implies that there could be extenuating factors (e.g. rain, if the job is outside) that could change the duration of the job but, we assume that such factors are not present. This also implies that there is an average rate at which the work is performed and that all the workers work at the same average rate. In real life some people work faster than others, some work quickly then take a short break, then go back to work. Prepositions In

Prepositions are particularly difficult for English language learners as they are often used idiomatically, and their meaning and use in a phrase is not obvious to non-native speakers. Prepositions such as “by,” “in,” “out of” etc are frequently used in math and can have multiple and different meanings depending upon the context. It is best to simply have students notice their usage in a particular context and practice saying them in unison as part of the phrase, rather than spend a lot of time on each one in isolation.

Conjunctions So

This word is used in many different ways in conversational English (“So, we were walking down the street...” or “So?”). In mathematics, so has a particular and important meaning - “it follows that,” indicating that there is a causal relationship between one set of facts and what the facts that will follow.

Declarative phrases There are

This is a common term in English, and in the language of math, but is not commonly used in many languages. Some languages have no verb “to be” at all. In addition, the word “there” has several meanings in English. There also sounds the same as several words in English but are spelled differently and mean different things than there.

Adjectives and adverbs (continued)

Handout 1-14CA certain construction job usually takes four workers six hours. Today, one worker called in sick, so there are only three workers. How long should it take them to do the job?




Conversational English versus the language of mathematics (continued)

The word construction is also not a commonly used words for some English language learners and needs to be explained with pictures and examples.Alternative wordings might be: “ a construction job”, “suppose a construction job …”

The language of mathematics is extremely terse and economical. Ellipses (leaving words out) are far more prevalent in the language of mathematics than they are in conversational English and in many kinds of writing, especially fictional writing.

There are several instances in this problem: four workers (to complete the job) (in) six hours. This phrasing is particularly difficult as it brings together two different facts and two different numbers in one phrase.

three workers ( to complete the same job that usually takes four workers to complete in six hours) This is another ellipse which is common in ratio and proportion problems.

Idiomatic expressions

“Called in sick” – meaning that a worker became sick and did not come to work. The term implies that a telephone or some other less technological method was used, but we don’t need to know that. Nor do we need to know that the worker is sick. All we need to know is that the worker did not come to work.

How long should it take? This is an idiomatic expression. The conditional case is not necessary here. How long will it take, or how long did it take, might be easier to understand.

Ellipses

Words not often encountered in conversational English

Handout 1-14DA certain construction job usually takes four workers six hours. Today, one worker called in sick, so there are only three workers. How long should it take them to do the job?



Mathematical difficulties (some possible responses):

There is an assumption in most problems like this that all the workers work at the same average rate. This is an essential characteristic that simplifies the problem—indeed without this assumption, it would not really be a math problem. A key difficulty is knowing when such an assumption is or is not justified.

The second assumption is that the workers are interchangeable and can all do whatever is needed to complete the job. For example, if only one worker is an electrician, and the job includes wiring, then the problem cannot be solved and makes no sense.

There are two critical mathematical ideas that result from this. First, the total number of “person-hours” to do the job is a constant, in this example, 24. Second, more workers => less time; fewer workers => more time. This is a classic type of problem called an inverse proportion: x y = C, where x and y are variables.

The easiest way to solve the problem is to make use of a concept called person-hours. Since it takes four workers six hours to do the job, the job requires at total of 24 (4 x 6) person-hours. In order for three workers to complete the same job they would each have to work 8 hours, so that the total work done is 8 x 3 = 24 person-hours. (Divide the total person-hours, 24 by the number of workers, 3, to get the number of hours, 8.)

A second approach is to figure out how long it would take one person to do the job. If 4 workers take 6 hours, it would take 1 worker 24 hours. Then it’s easy to see that two workers => 12 hours; 3 workers => 8 hours; and so forth. As one variable increases, the other decreases.

A much harder way to solve the problem involves finding the individual rates of each worker in fractions of a job. The four workers can each do ¼ of the total job in six hours. Therefore, in one hour each does 1/24 (1/4 divided by 6) of the total job. So when only three workers come to work, they can do 3/24 of the total job every hour. They complete another 3/24 of the job for every hour they work. The problem then becomes how many times do we need to add 3/24 to get a complete job, 24/24. Since 8 x 3 is 24, the job will take 8 hours to complete.

Handout 1-14EA certain construction job usually takes four workers six hours. Today, one worker called in sick, so there are only three workers. How long should it take them to do the job?



Cultural difficulties (some possible responses):

Some students may find it difficult to understand the problem in terms of an abstract “job”. They may find it easier to understand concrete examples of jobs. For instance, washing dishes in a restaurant, filling up test tubes and testing for antibodies, packing boxes, distributing supplies, preparing meals, planting seeds in a field, etc.

The notion that one worker is interchangeable with another is quite alien to many people. In many cultures and in many people’s experience, that would simply not make sense. Modern industrialized societies assume the standardization of work and create work situations to make sure that workers follow standard procedures and take set amounts of time to complete a task. The idea that any worker can take the place of any other worker, and that each would complete a task in the exact same time, is not common in many of the cultures from which students come.

The important concept of standardization or common rate or unit rate may be difficult for some students to grasp. Modern technological society is predicated on making many things standard or the same rate or dimension (standardized tests, standard shoe size, etc.) Those of us born into this culture may not realize how central the concept of standardization is in our lives and in our thinking, and how mathematics to a great extent is built upon it. This is especially interesting when we consider that this concept has only gained its central importance in the past two hundred years, and only in certain parts of the globe.

For example, the idea that three workers are supposed to do the job of four workers is another puzzling concept to some, no matter how much time is allotted.

The idea that someone can “call in sick” may not be permissible in some of the jobs students know about. That implies that they have a kind of job security that many workers do not have. This may be a puzzling concept for them.

Handout 1-15

Think back on some of the strategies you discussed earlier in the workshop as well as others you have used in your own teaching.

What supports do English language learners need in order to solve the problem without lowering its cognitive demand and without teliing them the answer?

Look back on the list of challenges. Now, consider different kinds of supports and scaffolds to address those challenges, such as:

concretekinestheticlinguisticgraphic organizers



Handout 1-16ATurning Points Tool Teaching High-Level Mathematics to English Language Learners in the Middle Grades


LINGUISTIC SCAFFOLDINGExplicitly teach phrases and words common to the particular math

concept you are introducing. In this lesson, the concept is inverse proportion.

Make sure students hear, understand and use the vocabulary words that are associated with the concept: “inverse proportion.” This term is not in the word problem itself, but it will help students remember what these kinds of problems look like on the page. So as they are role playing a job,you could use the term “inverse proportion” as in “We call this kind of relationship - the fewer the workers the greater the time to complete the job/the greater the number of hours that each person has to work - an “inverse proportion.”

Why does the problem say “a certain construction job”? It will help to provide examples of construction jobs and other jobs (pictures would help as well): laying a brick wall, painting the interior, installing dry wall, cooking a meal for a large group. Ask students to come up with some of their own examples and repeat “a certain hospital job” or “a certain garden project” until using the word certain in the context of a ratio problem feels natural to them.

Explain that a “certain” construction job is the term in “math language” that is used to say that it could be any of these jobs, or another one like them.

Help students understand that “usually” means “in normal circumstances”. Ask them to think of things that could make a job take longer (rain or snow for laying a brick wall or painting the outside of a house). “Usually” means those conditions do not apply when the team is doing the work, as they are rare and not the “usual” under which the job is done. It also means that the workers already know how to do the work, etc.

Scaffolding the problem for English language learners

Scaffolding: a process of ‘setting up’ the situation to make the child’s entry easy and successful and then gradually pulling back and handing the role to the child as he becomes skilled enough to manage it. (Bruner, 1983: 60)

GENERAL BEST PRACTICETeach the concept while you teach the language: That is the best

scaffolding you can provide to English language learners. Vocabulary and linguistic structures should never be taught in isolation, they must always be taught in the context of working with and understanding a concept. (See next pages for this kind of scaffolding.)




Practice “How long should it take?” by asking students “How long should it take for you to walk home from school?” To walk to the cafeteria? Rather than breaking down the idiom word by word, it’s easier to treat it as a whole and help students understand that it refers to a length of time.

Handout 1-16C



A certain construction job usually takes four workers six hours kitchen task any # employees any # minutes office project cooks days cleaning chefs weeks painting painters seconds

Today, one worker called in sick, so there are only three workers.Yesterday any # employee didnʼt come therefore only any # employees cook quit cooks chef got another job chefs painter painters For example:Today, two employees didnʼt come, therefore there are only four cooks.

How long should it take them to do the job? will finish task complete project

For example: How long will it take them to finish the project?

English language learners need to be able to read and understand word problems as they appear in textbooks and on standardized tests. The sentence frames below start out with the original problem. A certain construction job usually takes four workers six hours.

They then are presented with a number of choices, from which they can create their own problems. The more word problems they write, the better they will become familiar with the language and the syntax of this kind of problem as they appear in textbooks and tests.

For example:A certain kitchen project usually takes six cooks an hour.

Student created problem: A certain kitchen project usually takes six cooks an hour. Today, two employees didnʼt come, therefore there are only four cooks. How long should it take to finish the project?

Handout 1-16DTurning Points Tool Teaching High-Level Mathematics to English Language Learners in the Middle Grades


CONCRETE/KINESTHETIC SCAFFOLDING

By the end of the lesson, students need to understand that the key mathematical idea (inverse proportion), is one in which as one variable increases the related variable decreases. The product of the two variables remains constant, in this case, person-hours to complete the construction job.

A teacher can call up a few students to the front of the room. Suppose it takes these students 10 hours to do a job. Now call up another student. Will the job take more time or less? Call more students until the students agree that more workers => less time. Then ask students to sit down one by one. Repeat until they agree that fewer workers => more time.

Show pictures of people working together to complete a task. Ask students to describe what they see the people doing. Ask them to think about how you decide how many people should work on a job and how long it takes to complete that job. Have students write, draw or act out a similar problem but with different possible jobs, situations, people, time period, etc.

Have students do the work in as standard a way as possible so that each person takes the same amount of time to complete a task.

Ask them to think about how a manager uses such information to figure out how long a task or job should take and how many people should work together on a task in order to complete it.

Give them some concrete jobs and ask them to come up with suggestions for how long the job should take to complete and the number of people who should work on the job.

REPRESENTATIONAL SCAFFOLDINGUse manipulatives such as Cuisinaire rods or more life-like looking cutouts

to represent the workers, and a grid to represent the time needed to do the work. Ask students to work with pieces to figure out a way to solve the problem. Encourage them to try several different ways. Discuss the different strategies and solutions until students have a good understanding of inverse proportion problems.




Metacognitive Reflection

What did you learn from participating in the presentations and questioning? Why is that memorable?

What did you learn from creating the lesson plan for English language learners? from working with colleagues? Why is that memorable?

Handout 1-18Turning Points Tool Teaching High-Level Mathematics to English Language Learners in the Middle Grades





Notes from Video #2


Handout 1-19A (2 pages of transcript)Turning Points Tool Teaching High-Level Mathematics to English Language Learners in the Middle Grades

TRANSCRIPT

Note: T.O. assigns students to work in their groups to discuss Problem 2.3 Part A #4. She gave them an incomplete sentence to use in analyzing their work:

“When I know _______ and ___________ I can _____________.”

Tariq [reading from his own paper] When I know the single layer and the identical layer I can find the total number of volume.

Eddy: There’s more, right?

Tariq: Hold on.

Eddy: and the what?

Tariq. I mean the identical layer.

Eddy [Talking to himself as he writes] … and the identical layer I can …

[BREAK IN CLIP]

T.O. [to Eddy]. What do we have here? [reads from eddy’s paper]. When I know the single layer … that is a perfect answer! You ready to say that out loud up there?

Eddy. Yes.

T.O. [looking at Auis’ paper]. What do we have here? Who came up with this idea?

[Auis points to Tariq.]

T.O. [talking to Eddy and Auis] Do we agree with him? How come? Why does it make sense?

Eddy: um …

T.O. [pushes extra blocks away so a 2 x 4 array is clearly shown on Eddy’s desk.]

Handout 1-19B (continued)


T.O. Eight, OK … And how many layers did we need to make our box? … We had 80 total cubes, correct? So how many layers of 8 did we need?

Tariq. 10.

T.O. [to Eddy] How did we use 8 and 10 to get 80?

[BREAK IN CLIP]

T.O. [Standing in front of room speaking to class.] Tariq, I want you to read your sentence out. And I want everyone’s ears wide open to hear what he has to say, OK? We have some excellent answers from these groups in the middle here. I didn’t quite get to every group. But see if you have something that’s pretty close to what they have.

Tariq (reads from his own paper): When I know the single layer and the identical layer I can find the total number of volume.

T.O. What do we think. Read it one more time so we can kind of think as he’s saying (it) [to Tariq] … kind of slow.

Tariq (reads from his own paper a second time): When I know the single layer and the identical layer I can find the total number of volume.

[BREAK IN CLIP]T.O. Now what’s your question? Or do you want to add something?

Student in Corner (also an English language learner): What kind um -- what are – what are the layers that are -- what are the layers, the single layers and the identical layers that can … that can be used to find the volume?

T.O. (restates): What are the single layers and the identical layers?

Tariq: The single layer of cubes would be 8. And the identical layer in which we stack – we need to stack 8 – we need –the single layers are 8 -- to get the height is 10. So we stack 8 cubes, 8 layers of cubes, I mean 10 layers of 8 cubes and the total number of cubes would be 80. And you times it.

T.O. (to student in corner): Does that answer your question? Do you agree or disagree with what he said?

Student in corner: I agree with what he said.






Notes from Video #3







Notes from Video #4



Transcript from Video #4


[Students enter and each one takes a card as teacher (voice over) explains the activity.]

Teacher [speaking to students]. What I want you to do is, I want you to take your card and I want you to walk around the classroom and see who you fit with. Talk to a partner, talk to a person and see who fits best with your card.

[Cut to five students standing in a circle:]

S1 girl in green facing cameraS2. girl in orange, on rightS3. Boy in white sweatshirt with back to cameraS4. Girl in Black blouseS5: Girl with glasses

S1. What is an [inaudible] … how tall an object is.

S3. I need “length.” [moves away]

Students milling around looking at each other’s cards.

Teacher [voice over] They had to have a reasoning, using mathematical language or pattern language, to back up their reasoning.

S3: [facing camera, reading] When you compare two qualities[sic] it something [sic] expressed as a fraction.

Student at right: No, because she doesn’t have …

S1: [back to camera]: But it could be an example.

Teacher: [off screen]: When you think you’ve found where you belong I want you to make a little group so we can have different groups around the room. I’m going to give you about one more minute to do this.

S1 [holding three cards speaking to teacher]: Most of them all go together ‘cause they are all a figure and a picture.


Transcript from Video #4


S1: [holds three cards] we have to figure out a way how to put it together. Cause we already know what to say with this ... we’re gonna do ... [lines up cards on table as shown] …

Bug is a figure and a picture ... and we learned that a image is another name for a figure and a picture … and … [reads card] “the number used to multiply the lengths of a figure to stretch or shrink it to, to a similar figure” … it is a scale factor …

Like that? [reverses last two cards] or we could like call it that. [reverses cards again] ‘Cause this all already goes together.

S3: It sounds better like that.

S1: It sounds better like this. [original order]

Teacher: [speaking to students who are standing in a circle.] I want you to have one person speak from your group and share out your ideas with the class. We’re going to spend about two minutes doing this. We want to hear all of your smart thinking. OK? Who would like to go first?

S3: [standing with two girls] Me. We think “Lug” can fit in “What is a length?”. Because length meant how tall an object is. And “Lug” is the tallest one for the Worm family.

Teacher. Do you agree or disagree? [waits] Does anyone want to add on?

S1. When “lug,” when you multiply “lug” it gets wider, it gets bigger [gestures with hands] the coordinates.

Jen. Fabulous. [to another group] Sure. Who’s your spokesperson?

[Camera is on a group of four girls].

Girl in pink: Glorimar.

Jen. Glorimar.

Glorimar: We thought um, width means “How wide an object is,” because the first three letters are the same of “wide” and “width”. So we got a picture and then we thought that width was how wide the picture is. And “glug” is wide, and um, “zug” is wide too. So we put it all together.

Handout 1-23Best Practices Modeled in Workshop


• 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© Copyright 2009 Center for Collaborative Education/Turning Points

Handout 1-24

COMMITMENT: I will try out the ________________ best practices throughout my math lessons for the next few weeks.

I will collect the following evidence in order to determine how well it is working in improving the access my students have to learning high-level mathematics.

Kind of evidence _______________________


Date: ______________________________

Best instructional practice for English language learners on which I am working:

________________________________________________________

Content

Instruction

Using mathematical language appropriately and accurately

Using mathematical language appropriately and accurately

Other Other

Changes I have made in my practice

Changes I have noticed in my students’ depth of

understanding and engagement

Content

Instruction


© Enid Lee Consultants www.enidlee.com/modified by Center for Collaborative Education/Turning Points

http://www.enidlee.com/modified

http://www.enidlee.com/modified



PROTOCOLS USED IN

WORKSHOP 1*

DISCUSSION PROTOCOLS

Think/Write/Pair/Share

Final Word

Numbered Heads

Speak/Respond/Follow-up

Question/Respond

* Protocols originally developed by National School Reform Faculty

www.nsrfharmony.org and Project Zero www.pzharvard.edu

Modified for English language learners by Center for Collaborative Education

http://www.pz

http://www.pz



BEST PRACTICE: USE DISCUSSION PROTOCOLSDiscussion protocols are designed

•to provide clear guidelines and expectations for class discussions that are accessible to all students,

•to ensure that English language learners have the time they need to process what they hear and formulate their ideas in a new language,

•to make sure that all students, including English language learners, are equal contributors to the discussion as speakers and listeners, and

•to allow for alternative ways of thinking to be aired and evaluated.



BEST PRACTICE: USE PRIOR KNOWLEDGE AS A LINK TO NEW KNOWLEDGE•Prior knowledge gives English language learners entry points to every lesson

•All “big ideas” in mathematics can be linked to the prior knowledge of all English language learners

•Open ended, higher order thinking questions are excellent ways of accessing this knowledge


Think

Write✎Pair

Share

Think/Pair/ShareAdaptations for English Language Learners


Think/Write/Pair/ShareAdaptations for English Language Learners

Steps of Think-Write-Pair-Share

1. Have students sit in pairs. Ask all #1s to raise their hands, check to make sure every pair has a hand raised, then ask for all #2, check to make sure every pair has a hand raised.

2. Announce a rich discussion topic or problem to solve. (Example: Which room in our school is larger, the cafeteria or the gymnasium and how could we find out the answer? What does larger mean in this case?)

3. Give students at least 10 seconds of think time to THINK of their own answer. (Research shows that the quality of student responses goes up significantly when you allow "think time.") Give them another 30 seconds to WRITE their responses - long enough to write but not too long that the pace slows down. Encourage them to come up with several responses.

4. Ask students to PAIR with their partner to discuss the topic or solution. Direct all #1s that they have one full minute (adjust the time to the complexity of the question) to tell their partner the answer to the posed question, partner #2 listens without talking for the entire time allotted. #2 then has 30 minutes to tell #1 what s/he heard from them. Then #2 is given one minute to tell #1 how s/he answered the question. #1 listens quietly the entire time, recalling what s/he heard in the next 30 seconds.

5. #1 then responds to the comments of #2, using sentence stems such as “I agree because….” “I disagree because….” I wonder …..” “ Do you think that …?” “Could you explain ….. to me. That isn’t clear to me. “ #2 responds to #1’s comments, then has 30 seconds to ask her/his own questions to #1’s comments.

6. Finally, randomly call on a few students to SHARE their ideas with the class.

Simplified Steps

1. Announce a discussion topic, one that is meaty enough for a good, rich discussion.

2. Give THINK time. Give WRITING time.

3. Ask students to PAIR with an assigned partner to discuss their ideas. At times, PAIR English language learners with other English language learners so that they can participate no matter what their lalevel of English.

4. Call on students to SHARE their ideas with the class.



Overview

Think-Write-Pair-Share is one of the most common cooperative learning structures, it's one of the easiest to implement and it creates a culture of equity and inclusiveness in classrooms. It’s particularly useful for English language learners as it gives them a chance to try out their thinking and practice talking over an idea in a safe structure before they speak to the entire class. It can be used throughout a class, especially when only a few students raise their hands in response to a question or when it is clear that students need a chance to explore a topic before speaking in front of the entire class.

The structure is extremely versatile but is particularly suited for generating higher order thinking. The steps are simple, but it's important to follow the steps exactly to avoid the "groupwork" pitfalls. Sometimes teachers are tempted to skip some steps and just say, "Turn to a partner and talk it over." Read on to find out why those words don't necessarily result thoughtful responses from each student or an equitable classroom culture.

Before you ask students to use think/write/pair/share on their own, make sure you model it for students, telling them exactly what step you are modeling and asking them to discuss what they heard and saw during each step. Chart the steps as you model them. If this is the first time students have tried out the protocol, have them practice following the protocol on easy cognitive material before you use it to discuss new material.

For ELLs, provide appropriate pattern language and sentence starters for them to use during each step.

Hints and Management Ideas • Assign Partners - Be sure to assign discussion partners rather than

just saying "Turn to a partner and talk it over." When you don't assign partners, students frequently turn to the most popular student and leave the other person out. You may want to pair an English language learner with another English language learner or with a native speaker of English, depending upon the situation.

• Change Partners - Switch the discussion partners frequently. With students seated in teams, they can pair with the person beside them for one discussion and the person across from them for the next discussion.



• Monitor Discussions - Walk around and monitor the discussion stage. You will frequently hear misunderstandings that you can address during the whole-group that discussion that follows.

• Rallyrobin - If students have to list ideas in their discussion, ask them to take turns. (For example, if they are to name all the geometric shapes they see in the room, have them take turns naming the shapes. This allows for more equal participation.) The structure variation name is Rallyrobin (similar to Rallytable, but kids are talking instead of taking turns writing).

• Randomly Select Students - During the sharing stage at the end, call on students randomly. The first time you do this, expect them to be quite shocked! Most kids don't listen well, and all they know is what they said! If you keep using this strategy, they will learn to listen to their partner.

• Math Problem-Solving - Place a complex problem on the overhead Ask students to think about the steps they would use to solve the problem, but do not let them figure out the actual answer. Without telling the answer to the problem, have students discuss their strategies for solving the problem. Then let them work out the problem individually and compare answers.

Benefits of Think-Write-Pair-Share • Think-Write-Pair-Share is easy to use on the spur of the moment. • Providing "think time" increases quality of student responses. • Providing “writing time” gives all students time to develop responses

and encourages more thoughtful, fully developed responses.• Students become actively involved in thinking about the concepts

presented in the lesson. • Research tells us that we need time to mentally "chew over" new ideas

in order to store them in memory. When teachers present too much information all at once, much of that information is lost. If we give students time to "think-write-pair-share" throughout the lesson, more of the critical information is retained.

• When students talk over new ideas, they are forced to make sense of those new ideas in terms of their prior knowledge. Their misunderstandings about the topic are often revealed (and resolved) during this discussion stage.

• English language learners are more willing to participate since they don't feel the peer pressure involved in responding in front of the whole class.


Final WordAdaptations for English Language Learners

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 the part of the text they understand best.



#1 #2

#4 #3

➟

➟

➟

➟

#4

#1 #2

#3

➟

➟

➟

➟



1. Choose a text for students to read and give them time to read.2. Have everyone identify one significant idea or phase from the text. Make

sure it’s short - one or two sentences.3. Mark the phrase.4. Think about why you have chosen the phrase. It is a good idea to have a

“back up” phrase, a second choice in case someone else chooses your first choice.

5. Get into small groups and have each group choose a facilitator/timekeeper.

6. 1st person reads his/her phrase, making sure everyone knows where it is in the text and then has 1 minute (or any other short period of time) to say why that excerpt struck him or her. Why do they think it is important? Or why do they disagree with the excerpt or what important issues does it raise for him or her?

7. 2nd person says what s/he read 1st person say, responds to the excerpt briefly.

8. Next person in group does the same.9. Person #1, the person who chose the quote and started the discussion,

gets the “FINAL WORD.” S/he responds to what the others said and discusses what s/he thinks after hearing everyone else discuss the excerpt.

Continue until everyone gets a chance to share his/her quote, hear the others in the group respond, and then have the “FINAL WORD.”

Adaptations for English language learnersGive students sentence starters or pattern language to use in discussing their excerpt.I think _____ is important because...I heard you say that.... One connection I can make is... (See appendix for additional sentence starters.)

Adaptations for word problemsInstead of a text, choose a word problem for students to read. Each student can choose a phrase or fact in the word problem that s/he thinks is important for solving the problem.

Remind students to have more than one phrase or fact ready to discuss in case another student who goes before has already chosen that phrase.

Numbered Heads



Adaptations for English Language Learners

BEST PRACTICE: NUMBERED HEADS ASSESSMENT STRATEGY• encourages all students to hold each other accountable

• encourages all students to collaborate in creating a thoughtful and thorough response

• ensures that all students are given a chance to respond publicly

• efficient way to check for understanding with entire class and immediately address misconceptions and misunderstandings

Numbered Heads




#1 #2

#4 #3

➟

➟

➟

#1 #2

#3#4

➟

➟

➟

Small Group discussion

#4#4

Share out to class

#1#1

All #1s SHARE

All #4s SHARE

Numbered HeadsTurning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades



Overview

Numbered Heads is a simple and efficient strategy to check for student understanding throughout a lesson so that any misunderstandings or partial understandings can be immediately addressed. Since errors and misunderstandings can be flushed out and resolved as soon as they arise, this formative assessment strategy is a very good strategy for ensuring that all students reach mastery of critical mathematical concepts as quickly and as accurately as possible.

Numbered heads also establishes important cultural norms for the math classroom: It lets students know that every person in the class is capable of and will be held accountable for discussing their own answers and explanations to the class. It also sends the strong message to all students that they are all responsible for making sure that their partners or team members can adequately explain what the team or pair has discussed in their group. This supports the actively listening that they are required to do throughout the lesson as well as well as encourages them to ask each other good questions, to make sure that they check for understanding, and that they learn how to develop a good explanation for each response they give.

Numbered heads should be used after students have explored critical concepts using think/write/pair/share or other cooperative speaking, writing and listening strategies. For ELLs, it gives them an opportunity to rehearse their answers and gain a sense of confidence before they are required to speak in front of the class as a whole.

p.2

Steps

1. Students number off in teams, one through four.

2. Teacher asks a series of questions, one at a time.

3. Students discuss possible answers to each question in their teams for a set amount of time (30-90 seconds).

4. Teacher calls a number (1-4. All students with that number raise their hands, ready to respond.

5. Teacher randomly calls on students with specified number to answer on behalf of their team.

6. Teacher continues asking questions until the brainstorming or review session is finished.

Numbered HeadsTurning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades



Numbered heads does not guarantee, nor should it guarantee, that students be especially perceptive and accurate in their initial responses. Rather, it gives teachers a clear picture of what and how the students are thinking at the moment they are thinking it.

Once the teacher has that critical data on student understanding, s/he can ask “just the right question,” redirect the discussion through think/pair/share of other cooperative learning strategies, provide a new activity to address the misconception or error or any other type of scaffolding that will lead to more accurate and mature understanding by the students.

Since the strategy provides time for many different students to speak, it may seem as though it is too slow and cumbersome to use when pacing guides and other directives are the every day facts of life in today’s schools.

However, it paradoxically ensures that students move through the curriculum at a good speed since the strategy, and all other forms of formative assessment, make sure that when students are in the math classroom, they are actually learning important concepts. Time is not wasted later on in the unit or the year when teachers discover that students actually never really understood the subject matter and precious time is needed for review.

p.3

Speak/Respond/Question/RespondAdaptations for English Language Learners



BEST PRACTICE: RESPONSE/FOLLOW-UP QUESTION/RESPONSE• gives English language learners several opportunities to use the language of mathematics

• models a typical discussion for English language learners

• encourages students to go deeper in understanding the problem

• provides pattern language as entry points for English language learners

Speak/Respond/Question/RespondTurning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades


I notice that...

SPEAK

RESPOND

I agree because ....I disagree because...

➟What else do you notice?Why do you think that is important?

QUESTION

It is important because...

RESPOND

➟

➟

Turning Points Tool toTeaching High-Level Mathematics to English Language Learners in the Middle Grades


ADDITIONAL RESOURCES

• Visual Cues• Choosing a strategy to solve a problem•Giving Directions to English Language Learners• Using Pattern Language• Questions to Ask in Math Class• Sentence Starters to Use in Math Class• Examples of Organizing Templates/Graphic Organizers• Three Essential Lesson Planning Questions• Best Instructional Practices to Teach High-Level Mathematics to English Language Learners*• Answers to Frequently Asked Questions about Turning Points Tool• Additional Resources• References * see separate handout

Visual CuesTurning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades


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.

Visual CuesTurning Points ToolTeaching High-Level Mathematics to English Language Learners in the Middle Grades


Easy-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 a lesson.

Visual cues can be used 1) when giving directions to English language learners.

For example:

#4#4

All #4s SHARE

2) as a mnemonic device to remind English language learners of important discussion protocols

For example:

3) as a reminder of strategies English language learners can use to solve problems (see next page)

Draw a picture Make a list, table or chart

!

Use easier numbers Look for a pattern 1, 2, 3 10, 20, 30

Write an equation Work backwards a2 + b2 = c2


Visual Cues: Choosing a strategy to solve a problem


GIVING DIRECTIONS toENGLISH LANGUAGE LEARNERS


Giving Directions: Supports for ELL students

Provide a general overview of the process, then give the directions one at a time, and only just before the students are to do that particular part of the discussion.

Make sure the directions are clear and precise at each stage of the procedure. Reduce language load by using as few words as possible. Use body language. Write the steps on the board, emphasizing key words (both nouns and verbs.) Use symbols or pictures or different colored markers. Don’t crowd the board or the handout. Leave plenty of white space. Speak slowly and enunciate clearly. Model each step of the directions, then ask one or two students to model them for class. Check for

understanding from all students. Make absolutely sure that students have adequate time to think, plan and carry out the instruction. Use

one full minute, timed, as the rule of thumb, for the time needed to think and respond.

Set a purpose for speaking. If the student is a speaker, s/he should know exactly what she is supposed to speak about, what her audience should gain from her speaking, and what a good contribution sounds like. Model appropriate responses and ask students to list the characteristics of a good response. Check for understanding.

Set a purpose for listening. If the students are the listeners, they should be told exactly why they are listening and what will be expected of them after the speaker finishes. Each student must know exactly what s/he is supposed to do and how s/he will be held accountable for doing it well. Model appropriate responses and ask them to list the characteristics of a good response. Check for understanding.

If students are working in groups, design every role and every group activity so that each student is required to really think throughout the time they are in that role. Without that, students are not able to hold meaningful or extended discussions.

Encourage every student to actively participate and structure their roles so that they all must actively participate as speakers, as doers, and as listeners

Encourage risk taking and collaborative thinking. Tell them explicitly that any thoughtful idea counts as long as each person participates and they all thoughtfully and respectfully consider every response.

Give them the language structure to use in these kinds of discussions. “I agree with you because….” “I have another way of thinking about it…. It’s different because…., “etc.

Include a debrief!!! Make sure students are given time to reflect upon what worked and what didn't work, and how you and they can make it better next time. Providing a list of reflective questions helps them learn how to reflect and demonstrates the difference between telling what happened and reflecting on what happened.



BEST PRACTICE: SENTENCE FRAMES/PATTERN LANGUAGE•provide a clear scaffold for English language learners which they can use to participate in discussions

•are practiced with students before they are asked to use them individually

•allow students to have some choice in their responses

•encourage higher order thinking.

Here are some important steps to take in introducing pattern language (sentence starters/sentence frames) to English language learners.

Say the pattern language (sentence starter, phrase, question, etc.) Write the pattern language. Show students where the phrase appears in the handout.

Have students repeat the phrase in unison.

Model using the pattern language in several different contexts with which they are already familiar but never introduce it by itself. Always introduce it when it is necessary for the particular rich mathematical discovery on which the students are presently working.

Ask them to listen for clues about when this phrase is used.

Share those ideas with a partner.

Share them with the class/check for misunderstandings.

Model using it in a few situations when it is appropriate to be used, especially in any new situations or when introducing new mathematical contexts. Clarify when it is not used.

Ask them to try out the phrase with a partner, each person taking turns.

Ask one person from several pairs to say their phrases outloud/check for understanding.

Require them to use the phrase when it is appropriate, and check that it is being used appropriately.

************************************************************************************************

The following pages show examples of pattern language used in mathematics classrooms in the middle grades. Copy two or three phrases at a time for students, never more than a few.

Wait until they master each set, then introduce new ones. Mix up old and new phrases gradually until their repertoire of phrases becomes larger and larger and more and more automatic.

INTRODUCING PATTERN LANGUAGE


QUESTIONS to ASK in MATH CLASS when someone is explaining his/her answer

? ? ? ? ? ?Why do you think that? How do you know that is true?

Why do you think that strategy will work?

Can you explain that in a different way?

Do you see any patterns? Can you describe them?

Will this work with every number? Every similar situation?

What surprised you? Why did that surprise you?

Show us how you did that. Tell us what you did.

You are doing a good job of telling us what you did. Now, can you explain what you did, tell us why you did each step?

How do you know you have an accurate answer? How can you prove/justify it?

Will this strategy work every time you are trying to figure out a problem about ___________? When will it work? When won’t it work?

Can you use a different strategy to figure that out?

Can you give an example to help explain your answer?

What picture, diagram, chart or graph can you show to illustrate your solution?


SENTENCE STARTERS to USE in MATH CLASS when you are explaining your answer


“... .”

If ..., then...

The ____ is _____ since...

It seems to me that what this ____ really means is .....

It seems to me that what this _____ really is showing is that...

I believe that ...... Consider the following examples: ....

Since .... we know that ..... Also, .... Therefore, we can conclude that....

When computing ...... and ...., I noticed that both _____ had the same _______, while they had different _______.

We noticed that..... We know that..... Therefore, we can conclude that ...

If the _______ changes, then the _________ changes at the same ______.

If the_______ changes, then the _________ changes at a different______.

The number of _____ remains constant, while the number of _______ varies in the following way:...

QUESTIONS to ASK in MATH CLASS when someone is working on a problem

? ? ? ? ? ?

What do you think you know about this problem? What do you know for sure?

What questions do you have? What confuses or puzzles you?

Look at the problem carefully. Can you write down everything you notice that you think might be needed to solve the problem?

In what ways is this problem similar to others we have worked on? Look carefully and notice all the similarities.

In what ways is it different? Look carefully and notice all the differences.

What do you already know about the situation described in this problem? List all of the things you already know about the situation or the math that is involved.

What else do you notice? Keep looking/thinking.

What strategy will you try to solve this problem? What will be the first step?

What is the main question this problem is asking you to figure out?

Are there any words or phrases that are confusing to you, or used in a special way in this problem?

Can you draw a picture or create a chart or graph for this problem?

What patterns do you notice? What changes in the pattern and what stays the same?


SENTENCE STARTERS to USE in MATH CLASS when you are working on a problem

“... .”


I know that.... because....

One thing I don’t know is.... or One thing I’m not sure about is...

I notice that ..... I wonder/I think that might mean that....

This problem is similar to the ____ problem in the following ways:First,....Next,...Finally,...

This problem is different to the ____ problem in the following ways:First,....Next,...Finally,...

I am going to try the _______ strategy to solve this problem because... The first step I will do is to ...

Could you please repeat that?

Will you please explain that again?

Would you mind going over that again a little more slowly?

Can you explain that another way?

I think you are saying.... Is that right?

In other words, you are saying that …

I have a question about that. Do you mean...?

QUESTIONS TO ASK when you don’t understand or are not sure you understand

? ? ? ? ? ?


I agree with you because . . .

I disagree with you because . . .

I don't understand, please explain it in a different way.

In other words, you are saying that …

I noticed that...

That reminds me of...

I see a connection to...

You could also ....

I think what they mean is...

I have a question about...

RESPONSES when someone is giving their ideas



A certain construction job usually takes four workers six hours kitchen task any # employees any # minutes office project cooks days cleaning chefs weeks painting painters seconds

Today, one worker called in sick, so there are only three workers.Yesterday any # employee didnʼt come therefore only any # employees cook quit cooks chef got another job chefs painter painters For example:Today, two employees didnʼt come, therefore there are only four cooks.

How long should it take them to do the job? will finish task complete project

For example: How long will it take them to finish the project?

SCAFFOLDINGstudent responses/gaining fluency in reading and understanding the language of word problems

Student created problem: A certain kitchen project usually takes six cooks an hour. Today, two employees didnʼt come, therefore there are only four cooks. How long should it take to finish the project?

• Start with original problem (in bold). • Provide appropriate alternative choices. • Model how to make a similar but different problem, using the choices provided. • Have students orally practice creating several different choices until they gain fluency. • By doing this process with many different word problems, students will be able to recognize the way such problems use language in test situations and textbooks.

Draw a picture Make a list, table or chart

!

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




BEST PRACTICE: PROVIDING an ORGANIZING TEMPLATE/GRAPHIC ORGANIZER•saves time

•focuses English language learners’ attention on the mathematical concepts rather than copying in a new language

• models how to organize information

•provides clear way students can organize information

•creates expectations about # and quality of responses







Organizing Template/Graphic Organizer

Example: Double Entry Journal



Three Essential Lesson Planning Questions

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 ensure 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?

Answers to Frequently Asked Questions


Why does this tool focus on high-level mathematics for English language learners?Mastering high-level middle grades mathematics is critical to the life chances of all students. Middle grades students who do not understand and cannot apply the basic concepts of middle grades mathematics to a wide range of situations are at great risk of dropping out of high school, not attending college, and being unemployed or underemployed throughout their adult lives.

Far too many of these students are English language learners.

This professional development tool is designed to help math teachers in the middle grades provide English language learners with the opportunities all students deserve: access to and support in successfully learning high-level mathematics and the critical life skills and opportunities it provides.

How does this tool define “high-level mathematics in the middle grades”?Learning and mastering high-level mathematics in the middle grades, as defined in this tool, includes the following characteristics:

• Complex challenges, multi-step problems allowing for more than one solution path, different strategies, different entry points for different students

• Believable contexts that illustrate how mathematics is useful in everyday life, in science, in business and students’ communities and families

• Collaborative work with other students, where all students are accountable for what the group (small working group or the whole class) is learning

• “Meaty” problems that encourage discussing the reasons for different approaches, stating and clarifying ideas, developing academic language to describe the problem, solution strategies and results, and confronting gaps in one’s thinking

• Ways to organize information and look for patterns• Explaining – so other students can understand – how they solved (or are solving a problem) and how

they know when their answers make sense.

High-level mathematics in the middle grades emphasizes mathematics as a process in mathematical reasoning, a process in which students are engaged in understanding and accurately applying the critical concepts of middle school mathematics identified by the National Council of Teachers of Mathematics (NCTM).

High-level middle school mathematics problems are challenging for middle grades students. Students work in groups on problems that are slightly—but not too far—out of their comfort zone. Students make their own plans for how to proceed, and are required to explain their discoveries and their process through speaking and writing – and most important, to defend their methods and conclusions.

Another term used in this tool and by researchers (Stein, Smith, Henningsten and Silver [2000] ) to describe the work students do in such classrooms is “doing mathematics.” This term was coined to distinguish high-level mathematics from rote learning and/or following formulas to solve problems that research has shown is used in many classrooms throughout the United States, including those that enroll English language learners.

Answers to Frequently Asked Questions (continued)


How is “high-level mathematics” different from “mathematics” in the middle grades? Aren’t they the same?The goals listed above for “high-level” mathematics have always been among the goals of mathematics education, but “traditional” mathematics education has often put them in separate compartments, based on the premise that students must master basic skills and algorithms by rote before they can apply them to challenging problems. The result—most students learn to think of mathematics as a series of unrelated skills with no rhyme or reason, no connection to their own lives.

In addition, recent research has indicated that many teachers routinely lower the cognitive demand of problems as initially presented in textbooks, especially those specifically funded by the National Science Foundation to emphasize a constructive approach to learning mathematics (Smith and Stein). For example, teachers may tell students the steps they should take to solve a problem or give them algorithms or tricks to reach an answer without helping students develop their own understanding of how and why certain strategies work and what strategies are also possible.

What are the characteristics of teaching high-level mathematics to English language learners? Teachers who support English language learners, and all other students, to learn high-level mathematics are active through every step of the learning process; not by telling the students what to do or giving them the answers but by planning lessons that pose rich mathematical questions, by asking just the right questions, by listening carefully to their students’ responses, by redirecting their focus when misconceptions or misdirections surface, by presenting students with interesting challenges, by offering them a range of materials they can use to solve problems and by requiring students to explain their thinking throughout the lesson. Teachers encourage students to actively participate in mathematics lessons by introducing and practicing the language structures and vocabulary they need to do so and making sure they use them consistently throughout every lesson.

(Please see Best Instructional Practices for Teaching High-Level Mathematics to English Language Learners included in the Participant’s Packet to Workshop 1 and Facilitator’s Guide to the Workshop Series for a fully developed description of the characteristics.

How is this approach to teaching mathematics particular for middle grades students?Middle grades students seek challenges and a chance to express their own ideas. Teaching high-level mathematics to English language learners, and all other students in the middle grades, takes advantage of these key developmental traits of adolescence. Lessons that focus on high-level mathematics build in these traits by requiring all students • to choose a method for solving a problem, • to discuss these choices with peers, • to come up with workable and provable solutions that everyone can agree upon and everyone can explain with accuracy and clarity and • to apply this new knowledge to real life solutions that are meaningful to them and important to the community at large.



Aren’t these the same strategies and approaches that are successful with all students? Are there any strategies that are unique to English language learners?All middle grades students need the kind of support outlined above in order to learn the language of mathematics and master high-level mathematics. The strategies listed above can and should be used with middle grades students in general.

There are some challenges, however, that are unique to English language learners. Mastering middle grades mathematics means beginning to learn how to think as mathematicians. Thinking requires talking, listening, responding, reflecting and rethinking; it is by is nature language based. Middle grades students need support in mastering the language of mathematics, which is essentially a foreign language to almost every student who enters a math classroom.

Teachers, however, need to be especially vigilant in planning lessons for English language learners since these students are learning not just the language of mathematics but English as well. Teachers need to listen to and talk to English language learners in ways that take into consideration the specific and individual language and cultural needs of each language learner. They need to make sure that every English language learner learns how to express himself/herself so that they can actively participate in the lessons and develop the thinking strategies they need to master high-level mathematics.

This tool provides many different strategies that provide support so that a broad range of English language learners have access to and master high-level mathematics. This tool also provides information and encourages teachers to learn how to identify and access the needs of the individual English language learners in their own classrooms and design individualized support for them throughout their lessons.

(See the reading “Learning the Language of Mathematics in the Middle Grades” in the Participant’s Packet and the Facilitator’s Guide to Workshop 1 for a detailed discussion of the important language issues English language learners face in mathematics classrooms.)

Does this tool address the needs of English language learners who enter middle grades without mastery of basic computational skills?

English language learners, as well as native speakers of English, are very divergent in terms of their mastery of computational skills and mathematical facts. All of the strategies in this tool have been used in classrooms with English language learners entering with a great range of facility in mathematical and computational understanding and knowledge.

This tool focuses on providing all of these students with access to high-level mathematics in the middle grades, whether or not they are weak in computational skills and their knowledge of mathematical facts. This tool is designed therefore not to supplant support in these areas but to ensure that all English language learners are provided the opportunity to learn the high-level middle grades mathematics they need to move forward in school.



English language learners are a highly diverse group. Is access to high-level mathematics featured in this tool addressed to a particular group or level of English language acquisition?

This tool is designed to provide access to a broad range of English language learners from who enter the classroom in the beginning levels of English language acquisition to those in more advanced levels. The strategies demonstrate how to use the diversity of students’ background knowledge, language of origin, and ways in which they were taught mathematics.

The laws and regulations governing the teaching and assessment of English language learners differ from state to state. Is the tool designed to work in a particular state or group of states?

This professional development tool does not take a stand on any of the laws and regulations governing the teaching and assessment of English language learners. It can be used in a broad variety of classroom settings, including bilingual, ESL and SIOP settings.

Is the tool designed to be used with a specific textbook?

This tool, and the types of strategies is showcases and models, is designed to be used with a broad range of textbooks and has been used in settings which use different types of texts. The approaches modeled by the tool are effective in any middle grade mathematics classroom in which English language learners are enrolled.

Does this tool follow a particular model of training such as SIOP (Sheltered Instruction Operational Protocol) or CALLA (Cognitive Academic Language Learning Approach)?

This tool integrates the knowledge base and experience of a broad range of research based approaches successful in teaching English language learners, including SIOP and CALLA, as well as those that are effective in teaching high-level mathematics to middle grades students. Please see the reference and research section for further information.

Why is so much time devoted in the workshop series and in the model lessons to language based activities - talking, listening, writing and responding?

Middle grades students need careful and continuous support as they move from the basic computation and arithmetic of elementary school to the more complex topics of the middle grades: ratios and proportion, geometry, graphing and data analysis, functions and algebra. They also need explicit instruction in learning and using the language of mathematics; without facility in that language they are unable to discuss, process and understand the concepts of middle grades mathematics and beyond. Providing such support is especially critical for English language learners learning high-level mathematics.



Clearly, this requires that English language learners have multiple and repeated opportunities to hear, repeat, and try out their thinking - in a language that is new to them. All of the strategies in the workshop are designed to provide these opportunities in multiple ways so that English language learners have the critical support they need to succeed in mathematics in the middle grades.

Why are the directions for discussions so explicit and exact?

The workshop series makes extensive use of discussion protocols. These protocols are carefully structured to ensure that all English language learners become active and engaged participants throughout the math lesson. As in all protocols, it is essential to follow each step of the protocol with fidelity so that every student gains the depth of understanding built into each activity and has the time to contribute actively with other students. The facilitator guides the use of the protocol and adjusts the timing of each step to meet the needs of the group.

There is a far amount of writing in the workshops. What are the best ways teachers can encourage English language learners to write?

Writing is used in a number of ways throughout the workshops: 1) as writing to learn [THINK WRITE PAIR SHARE] and 2) as learning to write [Using sentence starters and pattern language to explain one’s thinking, ask questions and demonstrate one’s mathematical reasoning].

Both of these approaches help students master high-level thinking by giving them• time to think before answering,• an opportunity to consider various alternatives that might not occur to them initially but surface after a few moments after the question or activity is presented, • encouragement to provide more thoughtful and expanded ideas than a quick response would suggest, and • models for the kind of thinking required to do high-level mathematics in the middle grades.

It may appear counterintuitive to emphasize this practice with English language learners. We know, however, that English language learners need consistent and persistent modeling and support for mastering the academic language of mathematics in order to be successful students in middle school and beyond. Writing - with the support of sentence starters, pattern language and the gradual release of responsibility - provides this kind of support. The workshop gives English language learners the tools they need to participate in high-level mathematics in classroom discussions and succeed in standardized tests and other forms of assessment.

Additional Resources


•Adler, J. B. (2001) Teaching Mathematics in the Multilingual Classroom. Series: Mathematics Education Library, Vol. 26• Burns, M. (1990). A Collection of Math Lessons Grades 6‐8. Math Solutions• Burns, M. (1995). Writing in Math Class. Math Solutions• Burns, M. (2006) Marilyn Burns on the Language of Mathematics. Instructor Magazine•Coggins, D., Kraven, D., Coates, G.D. and Carroll, M.D. (2007) English language learners in the mathematics classroom. Corwin Press.•Evans, Linda. Building Background ‐ Bene\its of Using Sentence Frames to Build Background Knowledge. http://EzineArticles.com/?expert=Linda_L_Evans•Heinze, L. (2005). The Language of Mathematics. Presentation at TESOL Conference. http://kathrynheinze.efoliomn2.comhttp://kathrynheinze.efoliomn2.com

• Irujo, Suzanne. (November/December 2007). Putting it All Together: Integrating Academic Math Language into Math Teaching. The ELL Outlook (online resource)•Irujo, Suzanne. (March/April 2007). Teaching Math to English Language Learners: Can Research Help? The ELL Outlook (online resource)•Irujo, Suzanne. (May/June 2007) So Just What is the Academic Language of Mathematics? The ELL Outlook (online resource)•Moschkovich, J. N. Understanding the needs of Latino students in reform‐oriented mathematics classrooms. In L. Ortiz‐Franco, N. Hernandez, and Y. De La Cruz (Eds.), Changing the Faces of Mathematics (Vol. 4): Perspectives on Latinos. Reston, VA:NCTM, 5‐12.•National Council of La Raza (2005) Educating English Language Learners: Implementing Instructional Practices.•Slavit, D. and Ernst‐Slavit, G. (2007). Teaching Mathematics and English to English Language Learners Simultaneously. Middle School Journal, November 2007. Volume 39. Number 2. Pages 4‐11.•West Ed (2008). Helping English Language Learners acquire the language of mathematics. R&D Alert, Vol, 9, No.1, 2008.•www.ncela.gwu.edu/\iles/uploads/3/mathforELLs.pdf

Selected Resources

http://ezinearticles.com/?expert=Linda_L_Evans

http://ezinearticles.com/?expert=Linda_L_Evans

http://kathryn

http://kathryn

http://kathryn

http://kathryn

http://kathryn

http://kathryn

http://www.ncela.gwu.edu/files/uploads/3/mathforELLs.pd

http://www.ncela.gwu.edu/files/uploads/3/mathforELLs.pd

References


•August, S. (2006). Developing Literacy in Second‐Language Learners: Report of the National Literacy Panel on Language‐Minority Children and Youth. Center for Applied Linguistics.•Brown, C. (2005) Equity of Literacy‐Based Math Performance Assessments for English Language Learners. http://brj.asu.edu/content/vol29_no2/art5.pdf• Bruner, J. (1966) Process of Education. Harvard University Press.•Celeson‐Pattichis, S. (2009) Research Findings Involving English Language Learners and Implications for Mathematics Teachers. NCTM•Council of Great City Schools. (2003) English Language Learners in the Great City Schools: Survey Results on Students, Languages and Programs. •Dolan, S. (2009). Missing Out: Latino Students in America’s Schools. National Council of La Raza.• Francis. D. (2006). Practical Guidelines for the Education of English Language Learners. LEP Partnership Meeting Presentation.• Francis. D. and Rivera, M. (2006). Research‐Based Recommendations for Serving Adolescent Newcomers. Center of Instruction.•Heller, R. and Greenleaf, C. (2007). Literacy Instruction in the Content Areas. Alliance for Excellent Education.• Kieffer, M., Lesaux, N. & Snow. C. (2008) Promises and Pitfalls: Implications of No Child Left Behind for De\ining, Assessing, and Serving English Language Learners. The Regents of the University of California.•Learning, Institute for. High-Level Algebra and Geometry Tasks: Resources for Both Student and Teacher Learning. Connexions. 2 Apr. 2008 <http://cnx.org/content/m15958/1.2/>.•Moschkovich, J. N. Understanding the needs of Latino students in reform‐oriented mathematics classrooms. In L. Ortiz‐Franco, N. Hernandez, and Y. De La Cruz (Eds.), Changing the Faces of Mathematics (Vol. 4): Perspectives on Latinos. Reston, VA:NCTM, 5‐12.•National Council of Teachers of English. (2008). ELL Research Brief• Robertson, Kristina.(2009). Math Instruction for English Language Learners. http://www.colorincolorado.org.•Slavit, D. and Ernst‐Slavit, G. (2007). Teaching Mathematics and English to English Language Learners Simultaneously. Middle School Journal, November 2007. Volume 39. Number 2. Pages 4‐11.Stein, M.K., Smith, M.S., Henningsen, M., & Silver, E.A. (2000). Implementing Standards-based Mathematics Instruction: A Casebook for Professional Development. Second Edition. New York: Teachers College Press.Vialpando, J. and Yedlin, J. (2005) Educating English Language Learners: Implementing Instructional Practices. National Council of La Raza.Walqui, A.( 2006). Scaffolding instruction for English Language Learners: A Conceptual Framework. International Journal of Bilingual Education and Bilingualism 9(2),159-180.

Selected References

http://brj.asu.edu/content/vol29_no2/art5.pdf

http://brj.asu.edu/content/vol29_no2/art5.pdf

http://cnx.org/content/m15958/1.2/

http://cnx.org/content/m15958/1.2/

For additional informationcontact:

Dr. Sara Freedman, Project [email protected]


mailto:[email protected]

mailto:[email protected]

math for ell students workshop 1 participant handouts

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