Equity language for learners By Rusty Bresser, Kathy Melanese, and Christine Sphar Stefanie Timmermann/iStockphoto.com (2) Learn how to focus mathematical language on concepts to accommodate the needs of the 10 percent of U.S. students whose first language is not English. Copyright © 2009 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. E veryone uses language to learn mathematics. Paying close attention to the needs of students who are learning English as a second (or third) language is crucial so that we can modify lessons to accommodate those needs. The Equity Principle requires that we accommodate differences in our diverse student population to help everyone learn mathematics (NCTM 2000). About 10 percent of the nation’s public school students are English Language Learners (ELLs), and every year this percentage increases (NCELA 2007). Although language arts is probably the most challenging subject of their school day, mathematics is likely a struggle for these students as well. In fact, achievement data show that ELLs are not performing at the same levels in mathematics as their native English-speaking counterparts (NAEP 2007). Teaching ELLs Many educators share the misconception that because mathematics uses symbols, it is not associated with any language or culture and is ideal for facilitating the transition of recent immigrant students into English instruction www.nctm.org (Garrison 1997). On the contrary, language plays an important role in learning mathematics. We use language to explain mathematical concepts and carry out mathematical procedures. Students can deepen and cement their understanding of mathematics by using language to communicate and reflect on their ideas. When students converse about their mathematical thinking, such talk can help them improve their ability to reason logically (Chapin and Johnson 2000). Communication in mathematics has the potential to facilitate understanding, but the practice of discussing ideas in English may place children who are learning English as a second language at a distinct disadvantage, especially if they do not fully understand key vocabulary words, or if they have an incomplete understanding of syntax and grammar. The challenge of teaching mathematics to English language learners lies not only in making lessons comprehensible to students but also in ensuring that students have the language needed to understand instruction and express their grasp of mathematical concepts. ELLs have the dual task of learning the second language teaching children mathematics • October 2009 171 and the subject content simultaneously. For this reason, it is important to set both language and content objectives for ELLs. Just as language may not develop if we focus only on subject matter, content knowledge cannot grow if we focus only on learning the English language (Hill and Flynn 2006). Many teachers use supports such as manipulatives, visuals, and graphics to help students understand the content in their math lessons. These supports are important for building basic understanding of mathematical concepts but may not provide students with enough linguistic support for them to discuss their thinking, which would lead to deeper understanding of content. For example, if a student’s understanding of polygons is based on a two-column chart with drawings that distinguish polygons from shapes that are not polygons, once the chart is put away, the student may not have internalized enough of the lesson’s linguistic elements to be able to continue learning subsequent lessons on polygons. Having the language to talk about mathematical concepts is crucial to developing an understanding of those concepts (Bresser, Melanese, and Sphar 2008). The challenge for teachers is to focus on mathematical concepts and the academic language that is specific to mathematics. Teachers must be aware of the linguistic demands of their lessons and how they will address those demands explicitly during instruction so that ELLs can fully participate. During the lesson, will students be expected to do the following: • Compare and contrast numbers, data sets, or geometric figures. • Hypothesize while playing a number guessing game. • Describe the location of ordered pairs on a Cartesian coordinate plane. • Summarize a strategy they used to solve a multiplication problem. Each of these activities requires students to use language to communicate their mathematical thinking. And if students are expected to use these language functions in math class while they are learning English as a second language, they will need well-designed extra support. 172 October 2009 • teaching children mathematics Modifying math lessons During the past four years, we have collaborated on designing lessons to help all students, especially ELLs, accomplish two goals: develop English proficiency and develop mathematical understanding. Using existing mathematics lessons intended for native English speakers, we combined our knowledge and experience of English language development and of mathematics to modify lessons in ways that help us explicitly structure experiences to benefit all students, especially ELLs. When we modified lessons, we used the following questions to guide our work: 1. What is the lesson’s mathematical goal, and what language goal will support students’ understanding of this goal? 2. How can we support students when they are required to talk about their mathematical thinking? 3. How can we accommodate students with different levels of English language proficiency? 4. Are there vocabulary words with which students are unfamiliar, and how can we explicitly teach these words? 5. In what ways can we build in opportunities for discussion? Determining the language goal We based the language goals that we identified on the content of the mathematics lessons. After determining what we would expect students to do in each lesson, we analyzed the lesson for the language they would need to fully participate. This included vocabulary and language that teachers would use during instruction, as well as language that students would need to use to let us know if they had met our mathematical goals. For example, if students are working toward the goal of determining equivalent fractions, they need language to describe the fractions they create as well as language to compare and contrast them with other fractions. If the mathematics goal is that students solve a word problem using an effective strategy that makes sense to them, then they must be able to express, orally or in writing, the sequence of steps that they used to solve the problem. If students are learning about classifying and categorizing geometric shapes, the lanwww.nctm.org table 1 We used sentence frames for dice game predictions. Function Beginning Predicting The have Examples I will roll a six. will . Intermediate Advanced I predict that will . I predict that will because . I predict that I will roll a six. I predict that I will roll a six because I have rolled it more times than any other number. guage goal would be for students to describe the categories they chose for the shapes and to explain their reasoning. Each of these three mathematical goals— learning about fractions, solving word problems, and classifying and categorizing geometric shapes—has an accompanying logical language goal (or goals). In these instances, the respective language goals were to describe, compare and contrast, sequence, and categorize. With prompts such as the following from the teacher, native English-speaking children can perform these language functions to articulate their mathematical understanding: “Tell me what knowledge you have about fractions.” “Explain the steps you took to solve this word problem.” “Tell me what category this shape belongs to, and why.” ELLs may understand lesson content, but inexperience with the language can keep them from articulating what they know. Their struggles with the language of instruction can also lead them to partial or inaccurate understanding of the content. Until they verbalize their understanding, what they have or have not learned remains a mystery to the teacher and may even be unclear to students themselves. Choosing a language goal or language function that matches the mathematical content goal makes the learning more observable to all (Bresser et al. 2008). Once we determined the mathematical and language goals for a particular lesson, we designed sample ways that students could frame sentences to practice expressing their thinking about a mathematical concept with a partner, during small-group time, and during whole-class discussions. For example, after explicitly teaching key vocabulary terms such as predict, likely, unlikely, equally likely, and chance, we introduced sentence frames to help students make predictions (see Table 1) on the basis of data that they collected during a dice game. If students are sequencing the steps that they would use to solve a subtraction problem, they might use this sentence frame: Supporting classroom math talk Such frames can be powerful tools in the hands of students learning the English language. When structured appropriately, the frames are flexible enough to be used in a variety of contexts. They allow students to use key vocabulary terms to put together complete thoughts that can be connected, confirmed, rejected, revised, and understood. Just because we provide opportunities for ELLs to talk about their mathematical ideas in English does not necessarily mean that they will have the language to do so. Students must understand a lesson’s academic vocabulary, and they must also be able to use that vocabulary in their conversations during the lesson. www.nctm.org First I , then I . First I said, Eight, then I counted back three to get five. If students are comparing and contrasting quadrilaterals, they might use this frame: A has and , but a has both have . . A square has two pairs of parallel sides, but a trapezoid has one pair of parallel sides. A square and a trapezoid both have four sides. teaching children mathematics • October 2009 173 Accommodating various proficiencies The classrooms in which we worked included native English speakers and learners at different levels of English language acquisition: beginning, intermediate, and advanced. When modifying math lessons, we differentiated instruction by taking these levels into account and creating frames that were increasingly more sophisticated. While helping fourth graders compare circle circumferences and diameters, we decided that beginning-level students could talk about circumference and diameter using this frame: The is . The is Or students might say the following: The diameter is three centimeters. The circumference is nine centimeters.” We wanted intermediate-level students to use comparative language, so we modified the frames for them: The is than . The circumference is longer than the diameter. . Or they could use this frame: The diameter is short. The circumference is long. Reflect and discuss: “Equity for language learners” Reflect on the following questions related to “Equity for language learners” by Rusty Bresser, Kathy Melanese, and Christine Sphar. The prompts are provided as a tool to aid you in reflecting independently on the article, discussing it with your colleagues, and considering how the authors’ ideas might benefir your own clasroom practice. 1. What role does language play in learning mathematics? 2. List some challenges that English Language Learners (ELLs) face during math instruction. 3. State important points to remember when modifying a math lesson for ELLs. 4. How can teachers differentiate math instruction for ELLs with varying levels of proficiency in English? 5. What are the benefits of using sentence frames to teach mathematics? With the following questions in mind, consider a math lesson you plan to teach: 1. What is the lesson’s mathematical goal? 2. Which vocabulary terms must students understand and use? How can key vocabulary be explicitly taught? 3. What will students be able to say if they meet the math goal? In other words, state the terminology that students will use during the lesson. 4. For what purpose will students use language (e.g., to describe, to categorize, to hypothesize, to sequence, to compare and contrast)? 5. What strategies will you use to help ELLs understand mathematical content and generate language? 6. How will you differentiate the lesson for students whose English language proficiency levels vary? 7. Are opportunities for discussion built into the lesson? Tell us how you used “Reflect and discuss” as part of your professional development. Submit letters to Teaching Children Mathematics at [email protected] Include “readers exchange” in the subject line. Find more information at tcm.msubmit.net. 174 October 2009 • teaching children mathematics The is times than . The circumference is three times longer than the diameter. Advanced students and native English speakers need frames that help them make more complex statements: The of the is about/around/approximately . In addition to creating frames, we differentiated our questions to allow ELLs at all levels of English proficiency to respond and communicate their mathematical thinking. Students at beginning levels need simple prompts and questions that do not require lengthy explanations. We gave them opportunities to use physical rather than verbal responses: • Show me the polygon. • Touch the triangle. Questions with a yes or no answer are also appropriate for beginning ELLs: “Is forty-five greater than sixty-seven?” “Is the square a polygon?” To provide support, teachers can build in answers when asking short-answer questions: • Which shape belongs to this group, the triangle or the square? • Is this the diameter or the circumference? • Do more people on our graph like vanilla ice cream or chocolate ice cream? www.nctm.org table 2 Before students can use a sentence frame, they must learn key vocabulary terms that they can place in the frame to begin to talk about their learning. Function Describing Beginning A is/has . Examples A rhombus has four angles. Intermediate A is/has , and Advanced , A might have or but it will always have . . A square has four sides, four vertices, and no curves. Students with intermediate and advanced proficiency levels need less support to understand and respond to questions from the teacher, but carefully designed questions can elicit responses that reveal students’ mathematical thinking and allow for further development of academic English. For example, instead of asking an intermediate-level student, “What can you tell me about the graph?” you might phrase your question this way: “What do the most (or the fewest) people on the graph prefer?” This second question structure models an answer using academic discourse: “Most people prefer to swim. The fewest number of people prefer to ride bikes.” Compare this answer to the likely response to the first question: “A lot of people like to swim.” Students with advanced fluency can respond to even more open-ended questions and prompts such as, “Describe to me the steps you used to solve the problem and explain how you used them.” Teaching key vocabulary , A polygon might have three sides or six sides, but it will always have straight sides. Using sentence frames in primary grades In the primary grades, literate children are able to read different sentence frames and choose among them independently. Students with emergent literacy levels also make use of the frames, but in an auditory manner. As the teacher uses a frame, students hear how to structure their verbal responses. The teacher writes the frame or frames on the board to make sure she incorporates and models the vocabulary and syntax that will help students articulate their thinking. When the teacher introduces and models the frame aloud, it helps language learners to focus on the key vocabulary and learn how to structure some words around the key vocabulary in order to say something about the concept. This very careful and intentional use of speech by the teacher is what promotes oral language development, an important building block for literacy. For a lesson in which second graders had to sort and categorize objects by their attributes, Before students are asked to use a sentence frame, they learn key vocabulary terms that they can place in the frame to begin to talk about their learning. For instance, during a third-grade lesson on polygons, we explicitly taught key mathematical terms such as polygon, sides, vertex, vertices, open, closed, square, and triangle in the context of familiar objects in the classroom (see table 2). Then we introduced the sentence frames and had students practice describing familiar objects: “A calendar has four sides.” “A piece of paper has four sides, four vertices, and is a closed shape.” After practicing the vocabulary words and sentence frames in familiar contexts, students were ready to use academic language to learn about polygons. www.nctm.org teaching children mathematics • October 2009 175 • Meet the mathematical goal. • Meet the language goal. • Participate more than you normally do in math lessons. • Respond as if our sentence frames were appropriate for the lesson (not as if we used too many or too few frames). we first introduced names of objects (e.g., blocks, metal washers, rubber bands, crayons) that we wanted students to sort. Then we brainstormed with the students to identify some attributes they might use to categorize the objects (see table 3). Whether designing sentence frames for primary- or intermediate-level students, we had to keep in mind our chosen language goal or function that supported the mathematical goal. We molded the frames to fit the grade level and the particular lesson, and we considered the number of blanks necessary for each frame. We also had to think about what tense we wanted students to articulate their thinking in: past, present, future, or conditional. After creating the frames, we practiced using them to see if they made sense. The most important idea that guided us when creating sentence frames was to keep them as open and flexible as possible to allow students to express their own mathematical thinking. Modeling the sentence frames for students and giving them time to practice the frames were keys to students’ success. Reflecting on lesson modifications ta ble 3 After modifying and then teaching each math lesson, we took time to reflect on how successful the lesson was and what changes could make it more successful. We wanted ELLs in the class to do the following: 176 Guided by the idea that true reflection leads to action (Freire 1970), we used the questions as a way to assess the effectiveness of our teaching so that we could make specific changes that would benefit all students. Our goal was to level the playing field so that everyone had equal access to the mathematical content being taught. Modifying lessons—to make them comprehensible and to provide language support to help English learners think about new concepts, experiment with their knowledge, and solidify their understanding—is not easy. This is the type of work, however, that promotes equity and helps all students, especially English language learners, to fully participate in their learning community and fully benefit from our teaching. References Bresser, Rusty, Kathy Melanese, and Christine Sphar. Supporting English Language Learners in Math Class, Grades K–2. Sausalito, CA: Math Solutions Publications, 2008. ———. Supporting English Language Learners in Math Class, Grades 3–5. Sausalito, CA: Math Solutions Publications, 2008. Chapin, Suzanne H., and Art Johnson. Math Matters: Understanding the Math You Teach. Sausalito, CA: Math Solutions Publications, 2000. Dutro, Susana, and Carrol Moran. “Rethinking Sentence frames can be used to identify some attributes and then sort objects. Function Beginning Intermediate/Advanced Categorizing Expect beginning ELLs to use singleword responses to describe how objects are sorted. These are ; these are These objects are These objects are . Examples Blue! Red! White! Not white! These are blue; these are red. These objects are straight. These objects are not straight. October 2009 • teaching children mathematics . www.nctm.org English Language Instruction: An Architectural Approach.” In English Learners: Reaching the Highest Levels of English Literacy, edited by Gilbert G. García. Newark: International Reading Association, 2003. Farrell, Thomas S. C. Reflective Practices in Action: 80 Reflection Breaks for Busy Teachers. Thousand Oaks, CA: Corwin Press, 2000. Freire, Paulo. Pedagogy of the Oppressed. New York: The Seabury Press, 1970. Garrison, Leslie. “Making the NCTM’s Standards Work for Emergent English Speakers.” Teaching Children Mathematics 4 (November 1997): 132–38. Hill, Jane D., and Kathleen M. Flynn. Classroom Instruction That Works with English Language Learners. Alexandria, VA: Association of Supervision and Curriculum Development, 2006. National Assessment of Educational Progress (NAEP). 2007. http://nationsreportcard.gov/ math_2007/m0015.asp. National Council of Teachers of Mathematics www.nctm.org (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. National Clearinghouse for English Language Acquisition (NCELA). 2007. www.ncela.gwu. edu/policy/states/reports/statedata/2005LEP/ GrowingLEP_0506.pdf. Rusty Bresser, [email protected], lectures and supervises teacher education in the Education Studies Program of the University of California at San Diego. Kathy Melanese, [email protected], is a Distinguished Bilingual Teacher in Residence in the Education Studies Program at the same university. Christine Sphar, [email protected] sdcoe.net, is a beginning teacher support administrator and consulting teacher in San Diego’s El Cajon Valley School District. teaching children mathematics • October 2009 177

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