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Create a new document and edit with others at the same time -- from your computer, phone or tablet. Get stuff done with or without an internet connection. Use Docs to edit Word files. Free from Google.

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Engage NY

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To move from the status quo to the win-win solution, mathematics teachers of ELs have particular challenges to overcome, especially two commonly held opinions. First, there is a widespread belief that students need English proficiency in order to do mathematics reasoning tasks. A related conviction holds that the best way to create access to mathematics tasks for ELs is to lighten the cognitive demand of the tasks. Neither the "English first" nor the "lighten the cognitive demand" strategies about mathematics tasks for English learners is necessary for building proficiency in mathematical reasoning. Instead, we suggest remembering that ELs are thinking when working on mathematics tasks, and by letting that thinking become more visible and audible, teachers can nurture the productive potential in that thinking, as well as help adjust any faulty or misinformed thinking.

This book is intended to be a resource for mathematics teachers whose students include ELs. Furthermore, because this book explores the roles that language plays in the learning of mathematics, we also believe it can be useful to all teachers of mathematics, regardless of the particular collection of students in their classrooms. But these strategies, tailored in ways to meet the needs of different students, are absolutely essential for ELs. To meet ELs' needs it is necessary to nurture mathematics teaching practices that "specifically address the language demands of students who are developing skill in listening, speaking, reading, and writing in a second language while learning mathematics" (Celedón-Pattichis and Ramirez 2012, 1). In other words, although ELs must gain facility in using English to express themselves mathematically, in order to succeed in mathematics, the learning can and should happen "while learning mathematics."

"Have you ever asked yourself: How do I give my English learners access to the concepts and the language of the mathematics they are learning? Mark, Johannah, and Jill provide us with a simple, elegant, and brilliant answer that debunks two myths that plague English learners' educational experience-that "English first" and "lowering the cognitive demand of tasks" are necessary for reasoning about mathematics. Any teacher using the robust framework presented here-along with the connected routines-can be sure that all his students, but most important his English learners, will develop a positive mathematical identity as doers of mathematics. The way these routines provide students access by using language as a tool for thinking makes the book an invaluable asset for teachers of English learners."
-Harold Asturias, Director of the Center for Mathematics Excellence and Equity, University of California, Berkeley
" Mathematical Thinking and Communication is based on over a decade of research and development focused on meeting the needs of middle school English Learners in mathematics. Driscoll, Nikula, and DePiper present engaging examples to illustrate how teachers can put research-based recommendations for teaching ELs in mathematics into practice. As a mathematics educator who studies English Learners in mathematics classrooms, I was most impressed with how the authors distilled their extensive work into practical, yet flexible, strategies and techniques that others can use. I look forward to using this resource in my teaching and professional development with middle school mathematics teachers."
-Bill Zahner, Assistant Professor, San Diego State University
"Unlike most professional development, Mathematical Thinking and Communication does practice what it preaches. It uses visual and structured methods for helping teachers create access for English learners throughout the book. This is a great resource not only for teachers of students learning a second language, but also special educators working with students with communication challenges."
-Andrew Gael, special education teacher, New York City, blogs at The Learning Kaleidoscope

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Starting from the perspective that English learners are capable of mathematical thinking. Mathematical Thinking and Communication outlines four principles

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Supporting ELs in Math Class a Multimedia Professional Resource

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Supporting ELs in Math Class a Multimedia Professional Resource

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My dad gave me one dollar bill
'Cause I'm his smartest son,
And I swapped it for two shiny quarters
'Cause two is more than one!

- Excerpt from "Smart," by Shel Silverstein

As a student, I struggled with math. I didn't understand why it came so naturally to some students, but not to me. Looking back, however, I realize that I had an advantage that I wasn't even aware of - I understood the language in which the problems were written, even if I didn't understand how to solve them! Although it is easy to assume that many English language learners (ELLs) will excel in math because math is a "universal language" and students may have had prior educational experience that included mathematical instruction, that assumption can lead educators astray.

As I spoke with teachers and did research for this article, it became very clear that making sure that students understand math vocabulary and have ample opportunities to use it are very important. Solving word problems, following instructions, understanding and using mathematical vocabulary correctly - all of these skills require a language proficiency that sometimes exceeds our expectations. We tend to think of mathematics as a subject that does not require a strong command of language. In reality, however, mathematical reasoning and problem solving are closely linked to language and rely upon a firm understanding of basic math vocabulary (Dale & Cuevas, 1992; Jarret, 1999).

For many educators, the challenge of bringing language and math instruction together is a relatively new one. ELL teachers who hadn't taught content areas previously are now being asked to lead or support instruction in the math classroom, and many math teachers who don't see themselves as language instructors are now responsible for providing effective math instruction to ELLs.

High school math teacher Hillary Hansen learned just how big a role language plays in math instruction when she taught her first Basic Math course for ELLs last year. She wanted so much to provide the students with the good foundation they needed, but she felt unable to reach the students or engage them in her lessons, and by the end of the year she was exhausted and frustrated.

That summer she had an opportunity to join a district Sheltered Instruction Observation Protocol (SIOP) cohort to receive professional development and support to meet the needs of ELLs in content classes. She learned about the importance of language acquisition, building background knowledge, increasing student language production, and explicitly teaching academic language. She began this school year with a new set of tools and a deeper understanding of the instructional scaffolding ELLs need in order to learn the content while also learning English. I am happy to report that while Hillary still feels challenged and is working very hard, this year has been much more successful for her and her students.

As a result of more effective instruction, her students:

understand the content better and are working together to find creative ways to learn

discuss math more and know how to use the instructional supports their teacher has in place

are comfortable with math and asking questions to get the help they need.

Hillary feels that she is providing them with the foundation they need not only to understand the mathematical concepts, but also to successfully interact within a math classroom in order to continue learning more advanced concepts.

Following are some strategies that Hillary and some of the other teachers I spoke with found helpful this year, and that they recommend as best practices when teaching math to ELLs.

The importance of teaching academic vocabulary

Vocabulary instruction is essential to effective math instruction. Not only does it include teaching math-specific terms such as "percent" or "decimal," but it also includes understanding the difference between the mathematical definition of a word and other definitions of that word.

The following example, used in a presentation by Dr. Judit Moschkovich of the University of California at Santa Cruz, underscores why vocabulary must be introduced within the context of the content (Moschkovich, 2008):

In this problem, the student is instructed to "find x." The student obviously knew the meaning of the word "find" because he/she "found" it on the page and circled it. The student even put a note on the page to help the teacher in locating the lost "x". The student understood the meaning of "find" in one context, but not in the appropriate mathematical context.

I recently helped a math teacher create a Sheltered Lesson, and I was surprised to find that there were some vocabulary words that I didn't understand. My lack of familiarity with the words hindered my ability to do the math problem and gave me a deeper empathy for ELLs who struggle in the same way with vocabulary and comprehending math assignments. Following is a list of tips for explicitly teaching mathematical academic vocabulary:

Demonstrate that vocabulary can have multiple meanings. Help students understand the different meanings of words such as "table" and "quarter," as well as how to use them correctly in a mathematical context.

Encourage students to offer bilingual support to each other. Students will understand material better if they explain it to another student, and the new student will benefit from hearing the explanation in their first language. (Check the hotlinks for a list of bilingual translations of math vocabulary in multiple languages).

Provide visual cues, graphic representations, gestures, realia, and pictures. Offer students the chance to work with objects and images in order to master vocabulary. If there aren't enough items for each student, use manipulatives on the overhead or posted throughout the classroom, and demonstrate the vocabulary in front of the students. For example, Hillary created a Math Word Wall that has three parts: key vocabulary, "in your own words" definitions, and a variety of ways to portray a function. For example, multiplication is portrayed by the following symbols: x, *, and ( ).

Identify key phrases or new vocabulary to pre-teach. This strategy will help students decide which math function they should apply. Example: "more than" means "add." (See hotlinks for more references about math vocabulary.)

The importance of reading and understanding written math problems

Written word problems present a unique challenge to ELL students and teachers alike. In Reading and Understanding Written Math Problems, Brenda Krick-Morales writes, "Word problems in mathematics often pose a challenge because they require that students read and comprehend the text of the problem, identify the question that needs to be answered, and finally create and solve a numerical equation - ELLs who have had formal education in their home countries generally do not have mathematical difficulties; hence, their struggles begin when they encounter word problems in a second language that they have not yet mastered" (Bernardo, 2005).

Teacher Xiao-lin Yin-Croft has encountered this pattern in her classroom of bilingual Chinese students in San Francisco. She has developed a very creative way to use her students' background knowledge of math as a stepping stone for other language learning. She does this by accelerating math instruction at the beginning of the school year and then building on what students have learned in math in reading and other content areas. In Building Bridges for the Future, a Colorín Colorado "From the Heart" article, Xiao-lin explains her strategy:

First, we read math word problems; I demonstrate the logical thinking process while translating words into pictures and, finally, into number sentences. Soon, they start to explain their own thinking after reading complicated word problems that involve several steps. They correct each other, and argue about which number sentences they should use to arrive at the correct final results. As they sharpen their math skills, I capitalize on their enthusiasm to teach them how to extract the most important information from texts, and move them toward the oral and reading fluency they need to understand and discuss more challenging texts.

Even if you aren't accelerating math instruction, however, there are a number of ways to help students master word problems. Krick-Morales offers suggestions in the previously mentioned article, such as explicit instruction of key vocabulary, daily practice of problem solving, repeated readings of the word problem together as a class, and hands-on activities such as movement, experiments, or drawing to help students comprehend the problem. As students become more familiar with math vocabulary, they will be able to solve problems more easily.

The importance of building background knowledge

As the opening quote from the Shel Silverstein poem reminds us, background knowledge plays a critical role in math class! My colleague Hillary found that sometimes her students would get "lost" in a problem simply because they didn't understand the context. Following are some tips to help in building background knowledge of students.

Modify the linguistic complexity of language and rephrase math problems. Students will understand the problem better if it is stated in shorter sentences and in language they understand.

Guide students to cross out the unnecessary vocabulary in word problems. Doing so allows students to focus on the math function required. For example, one problem Hillary's students came across referred to a "school assembly." Even though the meaning of that phrase wasn't important in the solving of the math problem, students didn't know it wasn't important, and the lack of understanding contributed to their confusion.

Build knowledge from real world examples. Try to reinforce concepts with examples that students can picture and talk students through the situation. For example, if you need to paint a room, you need to know how much area will be covered so that you know how much paint to buy. Look for familiar ideas or props that can be used to engage students such as recipes, news stories about the economy, or discussion of personal spending habits.

Use manipulatives purposefully. This is important at all grade levels. Hillary has found math cubes to be very useful in having students represent the numbers in the problems and then manipulate the cubes to get the answer. She used the cubes and the terms "hot" and "cold" numbers when teaching with the concept of negative numbers. Students would use the red cubes as "hot" or positive numbers and the blue cubes as "cold" or negative numbers. As students laid out the number of hot cubes and cold cubes represented, they could easily see if the answer would be a positive or negative number by which color had the most cubes. A problem such as -2 + 1 = -1 would look like this:

The student then removed pairs of cubes - one red, one blue - until they could no longer remove any blocks. The remaining blocks represent the answer.

The importance of increasing student language production in the content area

As I've worked with content area teachers in my district to develop Sheltered Instruction lessons and activities to enhance ELL learning, I've told them, "If a student doesn't say it in your class, they're never going to say it." This is a bit dramatic, but it's true to some extent. When students learn new vocabulary, the opportunity to use it must be presented in class, because students are unlikely to try it out on their own - especially academic words like "parallelogram" or "function"!

Here are some tips to increase student-to-student interaction with academic language in the math classroom:

Have students translate symbols into words, and write the sentence out. Hillary used this strategy to check students' comprehension of problems before they solved them. For example, 3x + 4 = 16 would be written out, "Three times X plus four equals sixteen." This helps students process the operations involved in the question and gives them an opportunity to think through how to solve it. It also gives students a chance to familiarize themselves with important vocabulary words.

Create a "sentence frame" and post it on the board. Write the format of the sentence you would like students to use in discussion and then hold them accountable for using it. For example, "The answer is _______ degrees because it is a _________ triangle.

Have students share problem-solving strategies. This involves asking a simple question such as, "Did anyone else get the answer in a different way?" Then allow enough wait time so students can think through how their problem-solving process was similar or different to the one offered.

Allow students to discuss how they are thinking about math. This is a way of redirecting the lesson from teacher-to-student to student-to-student. For example a student might ask a question, "How do you know what kind of triangle it is?" Instead of the teacher answering and going to the board and pointing out the names and different triangles, the teacher can simply ask, "Does someone have an answer? Or "Would someone like to offer help to Mario?" Allow students to share how they think about the math concept and any tips they have for remembering the information.

Incorporate writing activities like math journals. This is an excellent way for students to process what they've learned and what questions they still have. The journal could start with simple prompts such as, "One thing I learned today..." "One thing I still don't understand..." "One way I can get the help I need..." "The answer to this problem is..." Writing out the answer to a problem is a very important skill to develop because many state math tests require a constructed response to questions.

Challenge students to create their own math problems. This can be a fun activity if students create a problem similar to the ones you have used in class and they exchange problems with a partner. By creating the problem and checking the answer they are reinforcing their own learning.

Using technology

Technology can also be a powerful tool in math instruction for ELLs. Here are some ideas ways you can play with technology in a math lesson:

Look for educational resources that accompany your school's technology tools and programs. There may be online or software programs available for teachers. For teachers who have an electronic whiteboard in the classroom, there are many resources available on links that can be easily accessed and brought into the classroom.

Look for interactive games that offer students a chance to practice their mathematical skills. Nintendo DS has an educational game called Brain Age, which is not language specific. The game provides excellent mathematical training for numbers and tracks results, showing student improvement over time.

Find out what's available online. Vital NY (Video Teaching and Learning for NYS Educators) on Teachers' Domain offers an online library of free media resources from the best in public television. Teachers' Domain resources include video and audio segments, Flash interactives, images, documents, lesson plans for teachers, and student-oriented activities. (Free registration required.)

If students are using a graphing calculator, ensure that they learn how to use it. Based on background and prior educational experience, students might not be familiar with how to use a calculator nor some of the more sophisticated models, such as the graphing calculator. Give students a chance to practice solving problems with their calculators once you have reviewed different functions. Texas Instruments offers numerous activities and product tutorials in their educational materials.

Even if it doesn't come easily at first, there are ways to get ELLs excited about math. By keeping their language skills and needs in mind when planning mathematical instruction (and by helping your colleagues do the same), you will be taking important steps in helping students master mathematical concepts and skills - and who knows? Your students may be the next generation of economists, rocket scientists, and math teachers just waiting for the tools they need!

Note: I'd like to thank my Minnesota colleagues, Hillary Hansen at Burnsville Senior High School and Kim Olson at Hidden Valley Elementary, as well as Xiao-lin Yin-Croft from San Francisco, for providing many of the helpful math teaching tips in this article. It is inspiring to know that there are talented, creative teachers who are always finding better ways to teach and are willing to share the knowledge.
Common Core Math and ELLs: Blog Posts
These posts from our Common Core blog highlight resources that can be used in Common Core math instruction for ELLs.
EngageNY: Common Core and Math
The EngageNY website provides materials focused on curricular examples, standards for mathematical practice, and other materials for professional development.
Discovery Education: Puzzlemaker
Puzzlemaker is a free puzzle generation tool for teachers, students and parents where users can create and print customized word search, criss-cross, math puzzles, and more using their own word lists.
Catherine Snow: Word Generation
Catherine Snow's new website provides information and resources for educators who would like to learn more about Word Generation and how it is implemented. Includes links to comprehensive academic word list that students need to master to comprehend academic content.
Dave's Math Tables: English/Spanish Dictionary
List of English math vocabulary words with the Spanish equivalent.
PBS Teachers: Math Lessons
PBS Teachers offers a database of multimedia math lessons and activities that can be searched by grade or topic.
SMART Notebook Lesson Activities
Browse lesson activities and materials for classrooms using SMART notebook software. Choose the country or region, and search by curriculum standards, subject and grade level.
Colorín Colorado Webcast: English Language Learners in Middle and High School
This webcast features Dr. Deborah Short and discusses effective instructional strategies for teaching English language learner students in middle and high school, such as the SIOP model.
Translating Word Problems
This is a great site for teachers in the elementary levels, as it provides a list of keywords you can teach your ELLs to look for as they read word problems. Also included are useful ideas and tricks to better prepare students to understand written math problems.

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Math Instruction for English Language Learners

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My dad gave me one dollar bill'Cause I'm his smartest son,And I swapped it for two shiny quarters'Cause two is more than one! - Excerpt from "Smart," by Shel Silverstein As a student, I struggled with math. I didn't understand why it came so naturally to some students, but not to me.

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ELLs Count on Language Supports (article)

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ELLs Count on Language Supports (article)

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Laura Varlas

At her district's newcomer center, English as a Second Language teacher Barbara Gottschalk would engage parents in a visual exercise: compare a math textbook from 20 years ago to the one in use today. The former, a familiar reproduction of the kind of math newcomers most likely experienced in their home countries, was defined by rows of computation. In the current textbook, however, words abound.

"You could tell just by looking at the two that even a student with a strong background in math will need some language support in a mainstream math class," says Gottschalk, who teaches at Susick Elementary in Troy, Michigan. A student with interrupted or limited schooling, she adds, will need even more language support.

With the Common Core State Standards and, especially, the skills identified in the Standards for Mathematical Practice, mathematics has increasingly emphasized conceptual understanding and reasoning, in addition to procedural know-how. "Language is the entirety of the mathematics classroom," says Megan Rowe, a math teacher at Borah High School in a linguistically diverse district outside Boise, Idaho. "From the language I use when I'm teaching to the language students use when they're reasoning among their peers-I don't know how you could ever teach mathematics now without focusing on language."

Double-Edged Disadvantage

Yet there's evidence across the education sector that many still assume English language learners (ELLs) don't need language support in math, and that belief can create a double-edged disadvantage for such students. ELLs placed in mainstream math courses with little to no language support are more likely to flounder under increased linguistic demands. However, when ELLs are tracked into remedial classes-and researcher Eduardo Mosqueda reports that this is a common and pernicious trend-it severely reduces the likelihood that they will eventually take advanced math courses.

"Once you start looking at math through the lens of language, you see linguistic demand everywhere," says San Diego State University professor Bill Zahner, a former high school math teacher. Without awareness and planning to support ELLs with that linguistic load, language will continue to be a gatekeeper to the advanced courses that lead to in-demand, well-paying careers in STEM fields. For Zahner and others, it's a question of equity: "How do we meet the needs of linguistically diverse students in math?"

Build a Baseline

When working with ELLs, it can be easy to mistake lags in language for gaps in math learning. "Just because they don't have the words doesn't mean they don't know how to do it," says Gottschalk. That's why middle school ESL teacher Beth Morrow's first rule of supporting ELLs in the math classroom is assess, don't assume. Establishing a baseline of where students are in terms of math knowledge and language proficiency helps teachers tailor the content and English supports ELLs receive.

Gottschalk uses a simple, language-free, calculation-based assessment to screen students' content knowledge. But even with pared-down assessments, cultural differences in math notation can make screening students challenging.

"Although there are some common rules that all mathematicians and scientists use, worldwide, they manifest themselves differently depending on the culture," says Cassie Eatmon, math chair at Herndon Middle School in Virginia. For example, long division is set up differently in the United States than in South American countries, and Arabic notation is written right to left. "Subtracting in Arabic looks wrong to us, but it's actually just flipped," explains Gottschalk.

Incorporate Semantic Supports

For newcomers, Eatmon recommends gradually increasing language demands. "Phrase questions that require them to produce a minimal amount of English at the beginning," she recommends. Choice questions (that require a yes/no, true/false, and ways for them to communicate nonverbally via a word card or hand signals) are a good place to start. As newcomers develop confidence with language skills, stretch expectations for discourse.

"Newcomers need special handling for a while," says Rita MacDonald, researcher at WIDA and the Wisconsin Center for Education Research (WCER). "However, students are newcomers for the short term; they are language learners for a long time."

For the long haul, ELLs will need different levels of support to engage in increasingly language-rich mathematics. Visual representations of concepts, word walls, defining word parts (prefix, root, suffix), sentence frames, and note-taking practice-common tools in language arts learning-can be adapted in order to make math-centered conversations more accessible to ELLs. For example, Eatmon recommends adapting the traditional think-pair-share processing strategy to think- write-pair-share so that students will have a script of their thoughts to refer to. "That reduces the burden on ELLs to think and talk [in a new language] at the same time," she explains.

Language in Use

Researchers at WCER have developed six specific moves teachers can make to strengthen all students' reasoning skills in mathematics, especially ELLs ( see Figure 1), as well as seven tactics students can apply when working with ideas in math ( see Figure 2). These strategies develop language through doing mathematics, what MacDonald calls a "language in use" approach. "It used to be a common belief that you had to learn language before you can do stuff. We don't believe that works," she says. "You really only learn language through shared experience, by doing something."

MacDonald recommends introducing and practicing the student moves one at a time and providing prompts and sentence frames to cue students as they clarify, restate, support, challenge, compare, explain, or build on an idea (see http://stem4els.wceruw.org/resources/Student-and-Teacher-moves.pdf for examples). The goal, says MacDonald, is to "get ELLs in the habit [of understanding] that there are so many things they can do when an idea is presented."

Practice with discourse-centered mathematics helps students build the precision that is the cornerstone of math fluency. "In regular conversation, we don't tend to be precise and specific with our language. In mathematics, you need to be, but you don't just jump to that disposition right away," explains MacDonald. You get there through experiences, like needing to explain precisely the constraints of an argument, of having to use a very specific label for something. Those experiences happen in classes where students are practicing math in all four language domains.

Designed for Discourse

In a language-rich, ELL-supporting math class, students learn math through listening, speaking, reading, and writing-the four domains of language. But many classes, not just math, tend to fixate on the listening domain. "Students get all kinds of practice listening," says Gottschalk. "But they can easily go through a whole day of school without ever having to utter a single word of academic English."

"If you set the goal that every single student has the opportunity to speak, every single day," says Rowe, "you start to realize how few opportunities there are-at least there were for me-for students to talk about math."

Rowe has refocused her teaching from "What's the math for today?" to "What's the language for today?" and says that allows her to structure learning that gets to a deeper level of math thinking. "My students aren't just robots that can regurgitate a process or steps," says Rowe. "To learn math, they have to be able to talk about it."

One of Rowe's favorite strategies for eliciting math discourse is the "partner round-up." This strategy can be used to talk through all sorts of mathematical operations. For example, if students are practicing adding fractions, each student receives a card with a different fraction on it. They then find a partner in the class and work together to successfully find the sum of their fractions. Once they do, they trade cards and find a new partner and repeat the process. "This could have easily been a worksheet-20 problems, practice adding fractions on your own," says Rowe. But when it's done as partner exercise, her students get to talk to 20 different peers and hear the different ways they think through and explain a problem. "You go from something that was content-based to language rich, and that makes a big difference especially for ELLs."

Gottschalk's school uses journaling extensively in math class. Language is modeled and practiced as students discuss, and then write about, a problem of the day, she says. Teachers can get creative with how they set up journal prompts. MacDonald suggests, "Here's my favorite wrong answer from the last class. Let's figure out why that was wrong."

Opportunities to develop language, through identifying relationships or making generalizations within sets of problems, are implicit in some of the materials teachers have, adds Zahner, but those opportunities are often not highlighted. Zahner is currently working with a team of 9th grade math teachers on a project funded by a $1 million grant from the National Science Foundation to design secondary mathematics classrooms focused on meeting the needs of ELLs. There's a real urgency, he argues, to provide teachers with resources and supports to unpack the curriculum so that opportunities for discourse and reasoning, not just right answers, are prioritized.

Weeding Wordy Problems

With language at the center of the modern math classroom, it's important to note that not all words are worthwhile in the service of math learning. In fact, there are a lot of nonessential words and distracting or confusing contexts littering math texts. "If you use an example that talks about a horse, it better be the animal and not the gymnastics equipment," Eatmon warns.

Zahner cites a textbook example of a word problem for calculating the slope of a roof. The problem talks about two- by six-inch sized pieces of wood that will be used as vertical supports for the roof. These pieces of wood are unfortunately written as "2×6's." The dimensions of the board are not actually relevant to the rest of the problem, says Zahner. But for a student from Mexico, this aspect of the problem was derailing. "His answers to every other problem were correct except for this one. He got stuck trying to figure out how to use '12,' which is two times six."

Rowe adds that sometimes her students struggle with names. "If there's a Mr. Brown in the problem, they'll wonder who that is and why his name is relevant to the problem." Rowe changes names to simple identifiers: man, woman, or person.

"When we see language complexity popping up, we have to ask, is this necessary to the mathematical development?," says Zahner. "If it's not, can we take it out or plan around it?"

On assessments, Rowe also provides students with a recording of all test questions-audio files saved to Dropbox or Google Drive and represented as a scannable QR code on students' test sheets. This helps because most ELLs' receptive or oral language proficiency is higher than their reading proficiency. Through school-provided devices or students' own phones, students hear Rowe read each question-sometimes word-for-word or paraphrased, with steps to get started, or breaking down the question in ways that they'll understand.

Context vs. Content

In addition to removing unnecessary language, Zahner advises teachers to think carefully about the number and types of contexts used to develop a mathematical concept. Although it may be helpful to use multiple contexts to show students that math concepts are all around them, don't make students master a multitude of scenarios just so they can access the math content. In one text, Zahner's team found 12 different problem contexts used to explain linear functions.

"If you've never driven on a toll road and a problem is asking you to calculate the per-mile toll, but you're stuck on what is a toll?, that's going to stop you from accessing the content," he says. Instead, Zahner suggests focusing on a few relevant, key contexts. One way to do that is to find out what students are interested in and build math problems with those contexts. For example, teachers he works with recently changed a problem about pumping water out of a pool to one about rates for downloading files to your phone. "When the teacher introduced this context, everyone was engaged because all kids have opinions about who's got the good network or good rates."

"Keep in the forefront of your mind, 'What do I want students to do?', and build your question around that," says Eatmon. Simplifying the language doesn't mean watering down the math. Keep expectations high that these kids can do the math you've taught them, she adds.

Equal Signs

Planning for language supports in math helps ELLs communicate their ideas and invites them into STEM, which can have a profound effect on their career and earning opportunities, says MacDonald. "It's this huge, epistemic equity issue."

"If we equitably involve ELLs in math-position them as idea-generators alongside their peers, instead of just passive listeners-their language is going to develop much more quickly because they'll be creating and negotiating meaning together," MacDonald adds.

Eatmon urges educators not to confuse a lack of language for a lack of intellect. With the right supports, it does not take long for their wheels to catch traction and take off, she says. Eatmon has seen ELLs make five to six years of growth over the course of one year. "It's not that they can't do it; it's that they've never been taught."

Figure 1

© 2017 The Board of Regents of the University of Wisconsin System. Used by permission. Stem4els.wceruw.org.

Figure 2

© 2017 The Board of Regents of the University of Wisconsin System. Used by permission. Stem4els.wceruw.org.

Laura Varlas is the managing editor of ASCD Express and a contributor to Education Update.

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Laura Varlas At her district's newcomer center, English as a Second Language teacher Barbara Gottschalk would engage parents in a visual exercise: compare a math textbook from 20 years ago to the one in use today. The former, a familiar reproduction of the kind of math newcomers most likely experienced in their home countries, was defined by rows of computation.

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My dad gave me one dollar bill
'Cause I'm his smartest son,
And I swapped it for two shiny quarters
'Cause two is more than one!

- Excerpt from "Smart," by Shel Silverstein

As a student, I struggled with math. I didn't understand why it came so naturally to some students, but not to me. Looking back, however, I realize that I had an advantage that I wasn't even aware of - I understood the language in which the problems were written, even if I didn't understand how to solve them! Although it is easy to assume that many English language learners (ELLs) will excel in math because math is a "universal language" and students may have had prior educational experience that included mathematical instruction, that assumption can lead educators astray.

As I spoke with teachers and did research for this article, it became very clear that making sure that students understand math vocabulary and have ample opportunities to use it are very important. Solving word problems, following instructions, understanding and using mathematical vocabulary correctly - all of these skills require a language proficiency that sometimes exceeds our expectations. We tend to think of mathematics as a subject that does not require a strong command of language. In reality, however, mathematical reasoning and problem solving are closely linked to language and rely upon a firm understanding of basic math vocabulary (Dale & Cuevas, 1992; Jarret, 1999).

For many educators, the challenge of bringing language and math instruction together is a relatively new one. ELL teachers who hadn't taught content areas previously are now being asked to lead or support instruction in the math classroom, and many math teachers who don't see themselves as language instructors are now responsible for providing effective math instruction to ELLs.

High school math teacher Hillary Hansen learned just how big a role language plays in math instruction when she taught her first Basic Math course for ELLs last year. She wanted so much to provide the students with the good foundation they needed, but she felt unable to reach the students or engage them in her lessons, and by the end of the year she was exhausted and frustrated.

That summer she had an opportunity to join a district Sheltered Instruction Observation Protocol (SIOP) cohort to receive professional development and support to meet the needs of ELLs in content classes. She learned about the importance of language acquisition, building background knowledge, increasing student language production, and explicitly teaching academic language. She began this school year with a new set of tools and a deeper understanding of the instructional scaffolding ELLs need in order to learn the content while also learning English. I am happy to report that while Hillary still feels challenged and is working very hard, this year has been much more successful for her and her students.

As a result of more effective instruction, her students:

understand the content better and are working together to find creative ways to learn

discuss math more and know how to use the instructional supports their teacher has in place

are comfortable with math and asking questions to get the help they need.

Hillary feels that she is providing them with the foundation they need not only to understand the mathematical concepts, but also to successfully interact within a math classroom in order to continue learning more advanced concepts.

Following are some strategies that Hillary and some of the other teachers I spoke with found helpful this year, and that they recommend as best practices when teaching math to ELLs.

The importance of teaching academic vocabulary

Vocabulary instruction is essential to effective math instruction. Not only does it include teaching math-specific terms such as "percent" or "decimal," but it also includes understanding the difference between the mathematical definition of a word and other definitions of that word.

The following example, used in a presentation by Dr. Judit Moschkovich of the University of California at Santa Cruz, underscores why vocabulary must be introduced within the context of the content (Moschkovich, 2008):

In this problem, the student is instructed to "find x." The student obviously knew the meaning of the word "find" because he/she "found" it on the page and circled it. The student even put a note on the page to help the teacher in locating the lost "x". The student understood the meaning of "find" in one context, but not in the appropriate mathematical context.

I recently helped a math teacher create a Sheltered Lesson, and I was surprised to find that there were some vocabulary words that I didn't understand. My lack of familiarity with the words hindered my ability to do the math problem and gave me a deeper empathy for ELLs who struggle in the same way with vocabulary and comprehending math assignments. Following is a list of tips for explicitly teaching mathematical academic vocabulary:

Demonstrate that vocabulary can have multiple meanings. Help students understand the different meanings of words such as "table" and "quarter," as well as how to use them correctly in a mathematical context.

Encourage students to offer bilingual support to each other. Students will understand material better if they explain it to another student, and the new student will benefit from hearing the explanation in their first language. (Check the hotlinks for a list of bilingual translations of math vocabulary in multiple languages).

Provide visual cues, graphic representations, gestures, realia, and pictures. Offer students the chance to work with objects and images in order to master vocabulary. If there aren't enough items for each student, use manipulatives on the overhead or posted throughout the classroom, and demonstrate the vocabulary in front of the students. For example, Hillary created a Math Word Wall that has three parts: key vocabulary, "in your own words" definitions, and a variety of ways to portray a function. For example, multiplication is portrayed by the following symbols: x, *, and ( ).

Identify key phrases or new vocabulary to pre-teach. This strategy will help students decide which math function they should apply. Example: "more than" means "add." (See hotlinks for more references about math vocabulary.)

The importance of reading and understanding written math problems

Written word problems present a unique challenge to ELL students and teachers alike. In Reading and Understanding Written Math Problems, Brenda Krick-Morales writes, "Word problems in mathematics often pose a challenge because they require that students read and comprehend the text of the problem, identify the question that needs to be answered, and finally create and solve a numerical equation - ELLs who have had formal education in their home countries generally do not have mathematical difficulties; hence, their struggles begin when they encounter word problems in a second language that they have not yet mastered" (Bernardo, 2005).

Teacher Xiao-lin Yin-Croft has encountered this pattern in her classroom of bilingual Chinese students in San Francisco. She has developed a very creative way to use her students' background knowledge of math as a stepping stone for other language learning. She does this by accelerating math instruction at the beginning of the school year and then building on what students have learned in math in reading and other content areas. In Building Bridges for the Future, a Colorín Colorado "From the Heart" article, Xiao-lin explains her strategy:

First, we read math word problems; I demonstrate the logical thinking process while translating words into pictures and, finally, into number sentences. Soon, they start to explain their own thinking after reading complicated word problems that involve several steps. They correct each other, and argue about which number sentences they should use to arrive at the correct final results. As they sharpen their math skills, I capitalize on their enthusiasm to teach them how to extract the most important information from texts, and move them toward the oral and reading fluency they need to understand and discuss more challenging texts.

Even if you aren't accelerating math instruction, however, there are a number of ways to help students master word problems. Krick-Morales offers suggestions in the previously mentioned article, such as explicit instruction of key vocabulary, daily practice of problem solving, repeated readings of the word problem together as a class, and hands-on activities such as movement, experiments, or drawing to help students comprehend the problem. As students become more familiar with math vocabulary, they will be able to solve problems more easily.

The importance of building background knowledge

As the opening quote from the Shel Silverstein poem reminds us, background knowledge plays a critical role in math class! My colleague Hillary found that sometimes her students would get "lost" in a problem simply because they didn't understand the context. Following are some tips to help in building background knowledge of students.

Modify the linguistic complexity of language and rephrase math problems. Students will understand the problem better if it is stated in shorter sentences and in language they understand.

Guide students to cross out the unnecessary vocabulary in word problems. Doing so allows students to focus on the math function required. For example, one problem Hillary's students came across referred to a "school assembly." Even though the meaning of that phrase wasn't important in the solving of the math problem, students didn't know it wasn't important, and the lack of understanding contributed to their confusion.

Build knowledge from real world examples. Try to reinforce concepts with examples that students can picture and talk students through the situation. For example, if you need to paint a room, you need to know how much area will be covered so that you know how much paint to buy. Look for familiar ideas or props that can be used to engage students such as recipes, news stories about the economy, or discussion of personal spending habits.

Use manipulatives purposefully. This is important at all grade levels. Hillary has found math cubes to be very useful in having students represent the numbers in the problems and then manipulate the cubes to get the answer. She used the cubes and the terms "hot" and "cold" numbers when teaching with the concept of negative numbers. Students would use the red cubes as "hot" or positive numbers and the blue cubes as "cold" or negative numbers. As students laid out the number of hot cubes and cold cubes represented, they could easily see if the answer would be a positive or negative number by which color had the most cubes. A problem such as -2 + 1 = -1 would look like this:

The student then removed pairs of cubes - one red, one blue - until they could no longer remove any blocks. The remaining blocks represent the answer.

The importance of increasing student language production in the content area

As I've worked with content area teachers in my district to develop Sheltered Instruction lessons and activities to enhance ELL learning, I've told them, "If a student doesn't say it in your class, they're never going to say it." This is a bit dramatic, but it's true to some extent. When students learn new vocabulary, the opportunity to use it must be presented in class, because students are unlikely to try it out on their own - especially academic words like "parallelogram" or "function"!

Here are some tips to increase student-to-student interaction with academic language in the math classroom:

Have students translate symbols into words, and write the sentence out. Hillary used this strategy to check students' comprehension of problems before they solved them. For example, 3x + 4 = 16 would be written out, "Three times X plus four equals sixteen." This helps students process the operations involved in the question and gives them an opportunity to think through how to solve it. It also gives students a chance to familiarize themselves with important vocabulary words.

Create a "sentence frame" and post it on the board. Write the format of the sentence you would like students to use in discussion and then hold them accountable for using it. For example, "The answer is _______ degrees because it is a _________ triangle.

Have students share problem-solving strategies. This involves asking a simple question such as, "Did anyone else get the answer in a different way?" Then allow enough wait time so students can think through how their problem-solving process was similar or different to the one offered.

Allow students to discuss how they are thinking about math. This is a way of redirecting the lesson from teacher-to-student to student-to-student. For example a student might ask a question, "How do you know what kind of triangle it is?" Instead of the teacher answering and going to the board and pointing out the names and different triangles, the teacher can simply ask, "Does someone have an answer? Or "Would someone like to offer help to Mario?" Allow students to share how they think about the math concept and any tips they have for remembering the information.

Incorporate writing activities like math journals. This is an excellent way for students to process what they've learned and what questions they still have. The journal could start with simple prompts such as, "One thing I learned today..." "One thing I still don't understand..." "One way I can get the help I need..." "The answer to this problem is..." Writing out the answer to a problem is a very important skill to develop because many state math tests require a constructed response to questions.

Challenge students to create their own math problems. This can be a fun activity if students create a problem similar to the ones you have used in class and they exchange problems with a partner. By creating the problem and checking the answer they are reinforcing their own learning.

Using technology

Technology can also be a powerful tool in math instruction for ELLs. Here are some ideas ways you can play with technology in a math lesson:

Look for educational resources that accompany your school's technology tools and programs. There may be online or software programs available for teachers. For teachers who have an electronic whiteboard in the classroom, there are many resources available on links that can be easily accessed and brought into the classroom.

Look for interactive games that offer students a chance to practice their mathematical skills. Nintendo DS has an educational game called Brain Age, which is not language specific. The game provides excellent mathematical training for numbers and tracks results, showing student improvement over time.

Find out what's available online. Vital NY (Video Teaching and Learning for NYS Educators) on Teachers' Domain offers an online library of free media resources from the best in public television. Teachers' Domain resources include video and audio segments, Flash interactives, images, documents, lesson plans for teachers, and student-oriented activities. (Free registration required.)

If students are using a graphing calculator, ensure that they learn how to use it. Based on background and prior educational experience, students might not be familiar with how to use a calculator nor some of the more sophisticated models, such as the graphing calculator. Give students a chance to practice solving problems with their calculators once you have reviewed different functions. Texas Instruments offers numerous activities and product tutorials in their educational materials.

Even if it doesn't come easily at first, there are ways to get ELLs excited about math. By keeping their language skills and needs in mind when planning mathematical instruction (and by helping your colleagues do the same), you will be taking important steps in helping students master mathematical concepts and skills - and who knows? Your students may be the next generation of economists, rocket scientists, and math teachers just waiting for the tools they need!

Note: I'd like to thank my Minnesota colleagues, Hillary Hansen at Burnsville Senior High School and Kim Olson at Hidden Valley Elementary, as well as Xiao-lin Yin-Croft from San Francisco, for providing many of the helpful math teaching tips in this article. It is inspiring to know that there are talented, creative teachers who are always finding better ways to teach and are willing to share the knowledge.
Common Core Math and ELLs: Blog Posts
These posts from our Common Core blog highlight resources that can be used in Common Core math instruction for ELLs.
EngageNY: Common Core and Math
The EngageNY website provides materials focused on curricular examples, standards for mathematical practice, and other materials for professional development.
Discovery Education: Puzzlemaker
Puzzlemaker is a free puzzle generation tool for teachers, students and parents where users can create and print customized word search, criss-cross, math puzzles, and more using their own word lists.
Catherine Snow: Word Generation
Catherine Snow's new website provides information and resources for educators who would like to learn more about Word Generation and how it is implemented. Includes links to comprehensive academic word list that students need to master to comprehend academic content.
Dave's Math Tables: English/Spanish Dictionary
List of English math vocabulary words with the Spanish equivalent.
PBS Teachers: Math Lessons
PBS Teachers offers a database of multimedia math lessons and activities that can be searched by grade or topic.
SMART Notebook Lesson Activities
Browse lesson activities and materials for classrooms using SMART notebook software. Choose the country or region, and search by curriculum standards, subject and grade level.
Colorín Colorado Webcast: English Language Learners in Middle and High School
This webcast features Dr. Deborah Short and discusses effective instructional strategies for teaching English language learner students in middle and high school, such as the SIOP model.
Translating Word Problems
This is a great site for teachers in the elementary levels, as it provides a list of keywords you can teach your ELLs to look for as they read word problems. Also included are useful ideas and tricks to better prepare students to understand written math problems.

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Math Instruction for English Language Learners

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My dad gave me one dollar bill'Cause I'm his smartest son,And I swapped it for two shiny quarters'Cause two is more than one! - Excerpt from "Smart," by Shel Silverstein As a student, I struggled with math. I didn't understand why it came so naturally to some students, but not to me.