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“Negativo por negativo me va dar un… POSITIvo”: Translanguaging as a Vehicle for Appropriation of Mathematical Meanings. (pp. 99-115) -- course reading

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“Negativo por negativo me va dar un… POSITIvo”: Translanguaging as a Vehicle for Appropriation of Mathematical Meanings. (pp. 99-115) -- course reading

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Math Education Aspects of Students Who Use Two Languages

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Math Education Aspects of Students Who Use Two Languages

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-study about when students switch languages when doing math and why
-found that some prefer using a specific language when doing calculations/arithmetic, but another language for math conversations
-switch languages swiftly, automatically -- doesn't interfere with math reasoning and understanding

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How Do Students Use Two Languages When Learning Mathematics? Using Two Languages during Computation Clip

Overall, strong evidence suggests that bilingualism does not impact mathematical reasoning or problem solving. Bilingual speakers sometimes switch languages when carrying out arithmetic computations. Bilingual adults may have a preferred language for carrying out arithmetic computation, which is usually the language of arithmetic instruction. Reported differences in calculation times between monolingual and bilingual adults were miniscule. Evidence suggests that switching languages for arithmetic computation does not affect the quality of conceptual reasoning. This language switching can be swift, highly automatic, and facilitate rather than inhibit solving word problems in the second language, providing the student's language proficiency is sufficient for understanding the text of the word problem.

How Do Students Use Two Languages When Learning Mathematics? Using Two Languages during Conversations (Code-Switching) Clip

The language ability and language choice of the person addressing a bilingual child are the most significant variables for the child's language choice. Young bilingual students (beyond age 5) speak as they are spoken to. In mathematics classrooms, children will also speak as they are spoken to, depending on the language ability and choice of the person addressing them. We cannot use someone's code-switching to reach conclusions about his or her language proficiency, ability to recall a word, knowledge of a particular technical term, mathematical reasoning, or mathematical proficiency. Research does not support a view of code-switching as a deficit itself or as a sign of deficiency in mathematical reasoning. We should not assume that bilingual students switch into their first language only because they are missing English vocabulary. Rather than viewing code-switching as a deficiency, instruction for bilingual mathematics learners should consider how this practice serves as a resource for communicating mathematically.

How Do Students Use Two Languages When Learning Mathematics? Does Using Two Languages Impact or Reflect Mathematical Reasoning? Clip

Overall, there is strong evidence suggesting that bilingualism does not impact mathematical reasoning or problem solving. What does research say about when, how, or why students switch from one language to another? How can research help us understand whether switching languages impacts or reflects mathematical reasoning?

In mathematics classrooms, students who are bilingual and/or learning English might use two languages during arithmetic computation. There is evidence that adult bilinguals sometimes switch languages when carrying out arithmetic computations and that adult bilinguals may have a preferred language for carrying out arithmetic computation, usually the language of arithmetic instruction. There is some evidence suggesting that switching languages for arithmetic computation does not affect the quality of conceptual reasoning. This language switching can be swift, highly automatic, and facilitate rather than inhibit solving word problems in the second language, providing the student's language proficiency is sufficient for understanding the text of the word problem. These findings suggest that classroom instruction should allow bilingual students to choose the language they prefer for arithmetic computation and ensure that all students understand the text of word problems.

Students might also use two languages during classroom conversations, what linguist call code-switching. In mathematics classrooms children will use one or another language depending on the language ability and choice of the person addressing them. We cannot use someone's code-switching to reach conclusions about their language proficiency, ability to recall a word, knowledge of a particular technical term, mathematical reasoning, or mathematical proficiency. Research does not support a view of code-switching as a deficit itself or as a sign of deficiency in mathematical reasoning. We should not assume that bilingual students switch into their first language only because they are missing English vocabulary. Rather than viewing code-switching as a deficiency, instruction for bilingual mathematics learners should consider how this practice serves as a resource for communicating mathematically.

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Math Education Aspects of Students Who Use Two Languages

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Overall, strong evidence suggests that bilingualism does not impact mathematical reasoning or problem solving. Bilingual speakers sometimes switch languages when carrying out arithmetic computations. Bilingual adults may have a preferred language for carrying out arithmetic computation, which is usually the language of arithmetic instruction. Reported differences in calculation times between monolingual and bilingual adults were miniscule.

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How to Engage Emergent Bilingual Students in Your Math Classroom

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How to Engage Emergent Bilingual Students in Your Math Classroom

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-include rich math tasks so students can see how math is useful instead of disjointed
-make it relatable
-reduce the amount of context switching (build on context that has already been established with students -- scaffolding)
-create a language-rich learning environment (support sense-making by amplifying use of language instead of simplifying it, allow for opportunities for students to communicate and refine their mathematical thinking, structure and support productive math conversations)

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Take a peek inside an engaged math classroom with me. Imagine students doing math, instead of passively listening to how to do it. Notice them interacting with a high-quality curriculum packed with rich math tasks and grade-appropriate materials that intentionally develop problem solving and reasoning skills. See them in productive struggle when trying to make sense of problems, looking for patterns, attempting different methods, finding solutions, and communicating with peers both verbally and in writing.

Does this scenario sound familiar? If you were thinking of NCTM's process standards, Common Core's Standards for Mathematical Practice, or perhaps the Five Strands of Mathematical Proficiency, you are on the right pathway. Furthermore, I'll hypothesize that you have recognized an engaged math classroom as one with a heavy emphasis on communication. We don't always think about communication as an important part of the math classroom, but it's critical, especially for emergent bilingual students. Let's examine the two specific things they need from you, their math teacher, for success: opportunities to engage in meaningful mathematics and a language-rich environment with appropriate supports.

Engage in meaningful mathematics

So what is meaningful mathematics, exactly? Meaningful mathematics centers on mathematical sense making and the communication of how everything comes together in a logical way. It includes rich math tasks that allow students to move beyond engaging with math as just a series of memorized procedures that are executed without much thought.

We don't always think about communication as an important part of the math classroom, but it's critical, especially for emergent bilingual students.

Activities that include rich math tasks allow students to see math skills as a useful means to understand a mathematical concept, rather than a series of disconnected processes and procedures. As my colleague Ted Coe recently explained, "Ideas in mathematics build on other ideas in a logical fashion. Nothing just 'pops up' or stands alone." The NRICH Project out of the University of Cambridge elaborates on the value of the deeper approach rich math tasks offer in their article "What is a mathematically rich task?": "Rich tasks allow the learner to 'get inside' the mathematics," they explain. "The resulting learning process is far more interesting, engaging and powerful; it is also far more likely to lead to a lasting assimilation of the material for use in both further mathematical study and the wider context of applications." Yet for many students, the thread in math isn't clear. And this disconnect, to borrow from Ted again, "is a consequence of a focus on 'ways of doing' mathematics while ignoring the 'ways of thinking' about the mathematics."

Now that we've defined meaningful mathematics, let's explore two specific ways to engage emergent bilingual students:

Make the math relatable. It's important to include opportunities for students to see themselves within a context or relate to the topic at hand. This can be as simple as inserting students' names into a context or changing the context altogether, so it more closely relates to a variety of cultures. One of the strategies I found effective with my emergent bilingual students was to translate one or two key words into their native language when engaging in a new topic.

Reduce the amount of context switching. When possible, build on context you've already explored with your students rather than introducing too many different details for them to understand and remember. For example, if you have a great lesson or series of lessons where each task is a new scenario, consider altering the tasks to align with a single scenario. There's no need for you to change the actual mathematics; just standardize the context. Your emergent bilingual students will be able to engage with the math more deeply because they won't be spending as much time trying to understand the scenario.

Provide a language-rich environment

Engaging and meaningful math tasks create opportunities for students to use language often in the math classroom, helping them to strengthen communication skills and, for kids who are learning English in particular, to practice a language that is newer to them.

All students should engage with rich math tasks, but such tasks present instructional challenges for students who are learning English. Emergent bilingual students should-and can-be held to high expectations regarding the mathematics and simultaneously receive supports with the language. Now, that is very easy to state but a little more challenging to enact. Collaboration between content and ESOL teachers plus leveraging translanguaging are two approaches to consider when making English accessible to the emergent bilingual students in your classroom. I also typically follow four research-based principles for promoting academic language put forth by Stanford University:

Support sense making by scaffolding tasks and amplifying language, instead of simplifying it

Provide students with abundant opportunities to communicate and refine their mathematical thinking

Structure and support productive mathematical conversations in a variety of formats

Provide intentional opportunities for students to be aware of their own mathematical thinking and how they use language to communicate it

Including rich math tasks and utilizing these four principles for classroom instruction will assist in elevating the success of emergent bilingual students in their study of mathematics.

An example

Let's look at a simple task from Illustrative Mathematics to demonstrate how meaningful math and language come together in the classroom to deeply engage students in learning. It's lesson 2.3 for seventh grade and is called "Making Bread Dough." With this example, you'll be able to see how each mathematics principle is driven by your actions during planning and instruction.

The first thing to note is that this task has enough text to potentially make it challenging for English learners. To scaffold instruction for them, I encourage you to use the language strategy Three Reads. It's a great way to help students make sense of the context of a math problem and understand what they are being asked to do before they get started. With this strategy, you'll read the math problem three times. The first read is for students to understand the context of the situation. During the second read, you'll ask students to identify quantities they feel are important to the context. And, finally, in the third read, you'll support your students in brainstorming solutions by thinking of different approaches to solving the problem. This third read provides an excellent opportunity for students to reflect on their ways of thinking and their peers' ways of thinking, too. This makes students more aware of thinking processes, how they communicate mathematically, and the reality that there are often many ways to tackle a math problem. Not only does the Three Reads provide appropriate scaffolding, but it also intentionally promotes discourse.

Mathematical discourse is an essential component in any math classroom, and one that requires careful planning because of the interconnectivity of the mathematics, academic language, and communication. One of the biggest challenges I had as a math teacher was understanding how to effectively include English learners because I knew they would benefit from participating in the discussion as well as listening to it. I found the key was to identify the conversation supports that matched the task and the language demands. For tasks like "Making Bread Dough," I often had students share their thinking with a partner first and then used a sequence of student work samples to facilitate discussions between students. When necessary, I added sentence frames, and sometimes modeled them, to support the proportional-relationships part of the conversation. Some examples include "_______ times as many as _______" or "_______ for every _______ of _______." This small addition would often be enough support to encourage English learners to voluntarily participate in the discourse.

Emergent bilingual students should-and can-be held to high expectations regarding the mathematics and simultaneously receive supports with the language.

There are several cultural and language considerations for this task as well. One way to invite students into the task is to ask them questions about their experience with baking bread and perhaps how bread is served in their culture. You can use visuals, like measuring cups and spoons, along with gestures while you demonstrate how to measure the ingredients. A linguistic consideration is recognizing that the word "table" has many different meanings by using visual examples for a table as a piece of furniture and as a graphic representation. In addition, it may be beneficial to point out the rows and columns in a table because this terminology could surface within the discussions.

Tying it all together with intention

As emergent bilingual students work to make sense of the mathematics, it is important to ensure they do not struggle with the associated academic language. When lessons have explicit goals for language, be sure to make your communication expectations of students clear and that you build your lessons with those goals as a guide. Take a look at the learning goals for the dough-making task below as an example. Notice the language goals in parentheses. When designing my lesson, I would focus on reminding myself that I must provide students the opportunity to speak and write about the proportional relationships if I want them to be successful.

The good news is that if the materials you use do not contain supports for your emergent bilingual students, you can make slight modifications and additions by keeping the four guiding principles I mentioned earlier in mind. I usually start by analyzing the mathematical tasks for language demands and then determining the necessary supports to meet those demands. Ask yourself some important questions: What are the language goals of this lesson? How will I provide students the opportunity to communicate their mathematical thinking (writing, speaking, listening, representing)? What support will students need to communicate effectively? How will they refine thinking? As you work through the answers to these questions in planning for instruction, the lesson will evolve and engaged emergent bilingual students in your classroom will flourish.

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How to engage the emergent bilingual students in your math classroom

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To help emergent bilingual students in your classroom, give them lots of opportunities to engage in meaningful mathematics and a language-rich environment.

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Math Instruction for English Language Learners

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Math Instruction for English Language Learners

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-many assume math class doesn't need a strong language proficiency, but untrue
-following instruction, using vocab, understanding word problems are very much tied to language proficiency
-spend more time on vocab and emphasize it's meaning/multiple meanings it possesses (truly UNDERSTAND it and how to use it)
-have visual cues to coincide with vocab/concepts
-build background knowledge (understand context of real world examples)
-increase student language production in content area (e.g. sentence frames, write out mathematical notation into words, share problem solving strategies with one another out loud, writing in math journals, create own word problems)
-utilize technology (graphing calculators, interactive games)

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Math Instruction for English Language Learners

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Several years ago, I met a high school math teacher named Hillary Hansen who had just taught her first Basic Math course for English language learners (ELLs). She wanted so much to provide the students with the good foundation they needed, but she felt unable to engage them in her lessons. 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 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 the following 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.While Hillary still felt challenged and was working hard, the second year was much more successful for her and her students. She felt she was providing them with the foundation they needed not only to understand the mathematical concepts, but also to successfully interact within a math classroom in order to learn more advanced concepts.As a result of more effective instruction, her students:Understood the content better and were working together to find creative ways to learnIncreased their participation in discussions and learned how to use instructional supports in the classFelt more comfortable with math and asking questions to get the help they neededYou can learn more about some strategies that Hillary and other teachers recommend for teaching ELLs math in the information below, along with reflection tips and ideas for getting to know your students’ math backgrounds.ELL Math Resource CollectionThis resource collection features several ideas for teaching math to ELLs.Reflection QuestionsHow much math instruction have your ELLs received in their prior schooling?How much do their levels of math background vary?What are some of their strengths related to math?What are some of their challenges?What are some practical ways your ELLs are using math in their daily lives?What kinds of strategies have supported their success or engagement in other areas?What are some ways that collaboration or co-teaching might support your ELLs?What are some leadership opportunities you might give ELLs in pair work or small group work?Learn About Students’ Math BackgroundsTeaching ELLs: Tips for Math TeachersPrint PDF >It’s important to remember that ELLs’ math backgrounds may vary significantly. Some students may have had consistent schooling where they developed strong math skills. Others may have limited or no formal schooling, but may have practical skills related to time, money, or other daily topics. If possible, find out whether students have attended school regularly and what kind of experience they have had with math instruction and with math in general. Look for ways to tie instruction to student interests (like sports) and responsibilities at home that involve math, such as helping with family finances, working in family businesses, and holding other jobs. For example, students from migrant farmworker families may be using math related to agriculture, such as weighing or selling produce. Also keep in mind that students may use other numeric and measurement systems students, such as metric system, Celsius, military time, etc.In addition, Pennsylvania educators Sally Lopez and Erin Minick also encourage educators to:Look for leadership opportunities for all students in math classProvide students with opportunities to “teach” others, work together, and demonstrate what they know in multiple waysGroup students intentionally and with sensitivity, avoiding the designation of specific students as “needing more help” in their grouping, partner assignments, or placement in the classroom(Ms. Lopez and Ms. Minick conducted a book study focused on Teaching Math to Multilingual Students, which you can learn more about from their website.)Keep in mind that veteran educators of ELLs also stress the importance of getting to know students as a key part of supporting student success. For example, in this NCELA webinar, Dr. Haiwen Chu, a national expert on mathematics education for English Learners at WestEd, recalls that visiting his high school students’ neighborhoods provided valuable insights on their experiences and familiar topics.See additional ideas for getting to you know your ELLs in 10 Strategies for Building Relationships with ELLs.The Language of MathAlthough it is easy to assume that many ELLs will excel in math class because math is a “universal language” and students may have had prior educational experience that included mathematical instruction, that assumption can lead educators astray. While computational skills may transfer, ELLs may still need support with language-focused tasks, such as solving word problems, learning (and using) math vocabulary, and writing explanations. 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), especially when considered along with the demands of college- and career-ready standards.Co-Teaching and CollaborationFor 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. When educators collaborate, however, they can better understand each other’s expertise and combine their expertise for the benefit of students.For example, when planning a lesson, a math teacher and an ELL teacher could collaborate to identify both the target content, such as key standards, concepts, or specific topics, as well as key vocabulary, academic language, and the language-focused tasks students will need to complete the lesson. These tasks might include writing out a proof, reading a math problem, explaining how to solve a problem, or writing out numbers as words. The ELL specialist could walk the math teacher through this process several times so that it becomes a more integrated part of the math teacher’s planning over time. After the lesson, the teachers could identify other areas of focus or clarification needed and share these insights with their colleagues.Another tool content teachers can use is language objectives, which help target the language demands of the lesson. You can learn more about them in the videos featuring Mr. Jesus Ortiz below.Teaching Academic VocabularyVocabulary instruction is essential for 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, as well as understanding the concept a vocabulary word describes.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 they “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.Teach key academic vocabularyAfter you identify your target content, identify the academic vocabulary words that students will need to understand that content. Some of these key phrases or new vocabulary words may also be helpful to pre-teach in small groups or mini lessons. Think about:Academic vocabulary (equilateral)Words and phrases needed to solve a problem (less than, given, arrange)Common words needed to solve word problems (train station, dolphin, favorite)Homophones (eight/ate, plane/plain, one/won, two/to/too, four/for, some/sum)Words that students are likely to encounter in other settings (formula, prove)Words that represent a numerical value necessary for solving the problem, such as measurement words (foot, feet, meter, pounds, kilometers)After you identify the vocabulary you wish to teach, use scaffolds for academic vocabulary, such as: Visuals: These can include visual representations of concepts, photos, and illustrations of key vocabulary words, posters, and number lines.Realia (real objects) and manipulatives: These might include hands-on objects related to a topic or small items, such as coins or small blocks for counting, sorting, weighing, etc.Graphic organizers: Graphic organizers are tools that allow students to organize information in a visual or hands-on way.Anchor charts: Anchor charts are posters that students and teachers co-create to highlight key concepts and vocabulary that students will need for reference.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 ( ).In addition, it can be helpful to ask students which strategies support their learning. Students often can provide valuable information on what is working for them!For examples of how vocabulary can impact students’ understanding of math problems, see the related video with ESL teacher Anne Formato below.Teach words with multiple meaningsDemonstrate that vocabulary can have multiple meanings. Help students understand the different meanings of words, as well as how to use them correctly in a mathematical context. This is also a good opportunity for using visuals and realia. Examples include:YardPlotTableProofAreaEvenPlanePlaceTimesBillQuarterChangeDraw on students’ home languagesUse cognates, which are related words in two languages. English and Spanish have many cognates within academic terms, like fraction/fracción.Use translated words or phrases strategically. For example, New York University has published bilingual glossaries for several topics.Allow (and encourage!) students to use their languages in group work to solve problems, brainstorm, and review material. 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.Increasing Student Language ProductionThe language demands of math are complex. Working with an ELL specialist can be a good starting point for identifying some of the math demands of the lessons you are teaching, as well as for planning how to teach and scaffold those language demands.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. 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.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. (Learn more in our sentence frame and sentence stems strategy.)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, such as entrance/exit tickets or math journals. This is an excellent way for students to process what they’ve learned and what questions they still have. The journal or exit/entrance ticket prompt 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…” “Something that helped me was...“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.Teach students how to read, interpret, and summarize data. Provide students with explicit, step-by-step instructions in how to read charts, graphs, tables, and infographics, as well as how to describe what they observe in visual representations of data.Model the kinds of writing students are expected to complete (e.g., written explanations). Students need to see and understand what they are expected to do and produce, such as explaining how they solved a math problem. This is a great opportunity for the “gradual release” strategy -- I do, we do, you do.Try AI tools. Consider using AI tools to rewrite math problems for students at early levels of proficiency or to generate math problems with familiar vocabulary that achieve your target objective. Bilingual Glossaries in Content AreasNew York University has published several bilingual glossaries in multiple languages that cover different content areas, including elementary math, middle school math, Algebra 1 and 2, Calculus, and related math terms.Reading and Understanding Written Math ProblemsWritten 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, 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.Identify key background knowledge in math problemsIt’s important to identify what kinds of background knowledge students will need to understand a math problem. For example, Ms. Lopez and Ms. Minick used this activity with a conference session: Ms. Lopez chose four math problems from her son’s math textbook and asked session attendees to brainstorm the kinds of background knowledge students might have questions about, some of which were relevant to solving the problem and some of which weren’t. Here are some things that attendees came up with:A math problem about a coupon and sales tax at a dog groomer: What is a dog groomer? Why do people take their dogs to get groomed? What is sales tax? What is a coupon? What is “deluxe”? What is an estimate and a discount?A math problem about people receiving a voucher for a free water bottle at a football game: Is football different from soccer? What is a voucher? Does the water bottle come with water inside it?A math problem about a fishing pole on sale: What is “percent off”? What is a sale? What is “regularly priced”? What steps have to be followed to solve this problem?A math problem about calculating a tip in a restaurant: What does “leave a tip” mean? What is a tip? Does the fact that there are four sisters (Emilia and three sisters) impact the outcome of the problem? Why aren’t they using an app like Venmo?As seen from the example above, this can be a powerful area in which math teachers and ELL teachers can collaborate.The Role of Background KnowledgeWhen it comes to background knowledge that impacts students’ understanding of new concepts, students may have relevant experiences to what you are teaching that just need to be drawn out (as in the vignette below). In other cases, students may not have the needed background knowledge. 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 tapping into and building background knowledge of students.Identify connections to your students’ experiences. When planning the lesson, consider whether they might have relevant experiences, and as you introduce the lesson, find out what they know, giving them an opportunity to make connections. They may come up with connections you never had imagined!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. This may be an area in which you wish to experiment with AI tools that can rewrite problems.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. Classroom vignette: A moment of humilityA middle school math teacher was trying to teach points on a grid and vertical/horizontal axis to her ESL students, three boys from Vietnam. She tried many approaches, including visuals, games, simulations, and manipulatives, and was just about to give up after very little progress. Suddenly, a student said, “Oh, I know — same same you drive a boat!” (“The same as you drive a boat!”) It turns out the child had helped his uncle use the stars to navigate a raft when they were fleeing Vietnam and trying to reach refugee camps in Thailand. He explained the concept to the other boys in Vietnamese with gestures and drawings on the board, who knew exactly what he was talking about. The teacher reflected, “I felt so humble. These boys knew a lot more about math than I had given them credit for. I had failed to activate their prior knowledge, and in the process had almost given up.“Source: This vignette is presented in Chapter 4 of Special Education Considerations for English Learners: Delivering a Continuum of Services. Using TechnologyTechnology can also be a powerful tool in math instruction for ELLs. Here are some 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.Find out what’s available online. Both PBS Learning Media and Vital NY (Video Teaching and Learning for NYS Educators) on Teachers’ Domain offers an online library of free public media resources. 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 calculator or other device, 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, graphing calculator, or math apps on devices such as phones or tablets. 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.Learn more about how to use AI tools in math. AI tools can be used to generate, differentiate, or rewrite a math problem; create a summary of a lesson plan; and many other tasks. Keep in mind that AI-generated content always need a careful review and edit. (See more in Using AI in Math from Edutopia.)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.Books About Math and MathematiciansIn addition, there are lots of books about math and mathematicians that you can incorporate into your classroom and classroom library. You can see a wide range of titles for different ages in our Math Resource Collection.Closing ThoughtsEven 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. You never know — 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.Related VideosHow vocabulary can get in the way of solving a word problemELL Specialist Anne Formato talks about the trouble a student had in understanding a written math problem based on a vocabulary word. Teacher Alejandra Rojas: Why cognates are your “best friend” in math classElementary teacher Alejandra Rojas, who teaches in a dual-immersion program, explains why cognates support vocabulary development in her Spanish-language math class. Teacher Alejandra Rojas: When bilingual students translate for their parentsAlejandra Rojas, a teacher at a dual-immersion school in Arlington, VA, shares some of her experiences as a child and teenager translating for her mother. How to Write Language Objectives: Tips for ELL EducatorsIn this video from Syracuse, NY, Jesus Ortiz, a bilingual teacher, learns how to write a language objective from Areli Schermerhorn, a peer evaluator with expertise in ELL and bilingual education. Teacher Observation Cycle: A Math Lesson on PictographsThis video highlights Mr. Ortiz’s math lesson on pictographs. While the main topic of the video is teacher evaluation, numerous best practices and strategies related to math instruction for ELLs are mentioned in the video. Recommended ResourcesNCELA Teaching Practice Brief: Math (A related podcast is available as well)Supporting English Learners in Math and Science: Effective Instructional Practices and Examples from NCELA’s Teaching Briefs PBS Learning Media: Math LessonsEngageNY: Common Core and MathCommon Core Math and ELLs: Blog PostsDiscovery Education: PuzzlemakerCatherine Snow: Word Generation SMART Notebook Lesson ActivitiesTranslating Word ProblemsColorín Colorado Webcast: English Language Learners in Middle and High School

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-be cognizant of the power of labels (have an effect on the students and how we view them, comes with assumptions and connotations)
-acknowledge fluid, dynamic language practice is the bilingual norm
-construct math classrooms rich in language across modalities -- speaking, listening, reading, writing, visuals, etc
-recognize the interplay between identity and language
-don't equate language proficiency with mathematical proficiency

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Translanguaging in the Mathematics Classroom: Engaging Emergent Bilinguals in Mathematical Literacies

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Translanguaging in the Mathematics Classroom: Engaging Emergent Bilinguals in Mathematical Literacies

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-discourse and academic language is important
-emphasis on dynamic bilingualism -- drawing from full language repertoire to make meaning

strategies:
-multilingual word walls
-cognate charts
-multilingual collab work (e.g. listen in english, discuss in any language)
-math language portfolio (journal to reflect on experiences with math specific language and track progress)

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THEORIZING A TRANSLANGUAGING STANCE: ENVISIONING AN EMPOWERING PARTICIPATORY MATHEMATICS EDUCATION JUNTOS CONEMERGENT BILINGUAL STUDENTS

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THEORIZING A TRANSLANGUAGING STANCE: ENVISIONING AN EMPOWERING PARTICIPATORY MATHEMATICS EDUCATION JUNTOS CONEMERGENT BILINGUAL STUDENTS

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-in bilingual classroom, students shared their thinking in either English or Spanish, teacher wrote their thinking on the board using mathematical thinking (same for each student) -- didn't translate, but allowed the math to serve as a means of translation across languages
-emphasis on familia -- student-centered discussion, responsibility of sense-making put on students with one another

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Translanguaging to Ensure Latinx Mathematics Learners Thrive

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Translanguaging to Ensure Latinx Mathematics Learners Thrive

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-after school math club designed to promote math bi-literacy
-played math games and problem solving (non-remedial)
-"students were encouraged to be self-directed, to work collaboratively, to verbalize their thinking, and to ask questions "
-all participants encouraged to speak Spanish
-"students communicated electronically with a mathematics wizard, El Maga, who engaged students in bilingual conversations about their mathematical experiences "
-children gravitated towards using English in academic spaces, but El Maga was persistent in using Spanish and encouraging biliteracy
-had a critical level of parent involvement, which helped bridge use of Spanish in family/community spaces to academic spaces

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Contesting math as the universal language: a longitudinal study of dual language bilingual education language allocation (access article through UC Berkeley)

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Contesting Math as the Universal Language

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Contesting Math as the Universal Language

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-purposeful clustering of students from DIFFERENT language groups in DLBE (dual language bilingual education) --> higher levels of bilingualism, cross-cultural reciprocity, and academic achievement
-bridging model (time between classes where there is a bridged switch between English and Spanish) --> math didn't use this b/c seen as universal
-dual language in math shown to be more effective than just ESL

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ABSTRACT
In this study, we used a quasi-experimental, longitudinal design to examine the English language arts (ELA) and mathematics test scores of emergent bilinguals (EBs) in DLBE 50:50 models versus English as a second language (ESL) models in Indiana over four academic school years. With a focus on the midwestern U.S., we bring attention to a less-studied area where the EB population is more recent, but quickly growing. Results show that students who attended a DLBE program rather than an ESL program, scored significantly higher on ELA tests, but the association between program type and mathematics outcomes was more complex. The discussion and implications caution the English as a second language, bilingual, and dual language bilingual education fields to conduct a comprehensive analysis of content allocation and content complexity in relationship to language designation, instead of solely focusing on the overall language allocation of the DLBE program model.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This study was made possible by funding from the US Department of Education's Office for English Language Acquisition: Grant #: T365Z170213.

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Contesting math as the universal language: a longitudinal study of dual language bilingual education language allocation

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ABSTRACT In this study, we used a quasi-experimental, longitudinal design to examine the English language arts (ELA) and mathematics test scores of emergent bilinguals (EBs) in DLBE 50:50 models versus English as a second language (ESL) models in Indiana over four academic school years.

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" Laura and Elise created a classroom environment where talking about language was as important as the mathematics lesson itself. "

-allowed use of any language in repertoire
-class followed a consistent routine (know what to expect)
-word problems in native languages with cultural context (photos on altar)
-teacher speaking in students' native tongues
-teacher repeats back what student said to clarify
-a lot of the time is spent talking about strategies and how a student thought (not just on solving math, but discussions using language)
-have visual models of what is being discussed