- Have you observed EBs working on mathematical word problems? What were they doing to solve the problem?
- Have you ever thought that certain mathematical tasks are not appropriate for EBs? If so, what type of task was it and why did you think it was not appropriate for EBs?
- Have you seen your EBs talking in their home languages? What did you do or what would you do if EBs used their non-English language in class or during recess?
The Dilemma with Word Problems
Consider the following vignette:
Ms. A is a 6th-grade math teacher. When her personal learning community (PLC) looked ahead at the unit assessment set for her next math unit, she sighed because it was full of word problems. In Ms. Smith’s 6th-grade math classes, she has about 30% Emergent Bilinguals, and about half of them are recent immigrants whose English proficiency is at the lowest level. Ms. Smith found some EBs to be very capable of mathematics and had been working hard to receive a good grade in mathematics. However, she was worried about whether they could demonstrate their mathematics knowledge properly with these word problems, many of which included complex sentences and difficult non-mathematical words.
What do you think about this story? Have you had a similar experience? Do you think this is common?
We agree that Emergent Bilinguals (EBs) can do mathematics. But, many teachers are likely to believe that word problems are difficult for EBs. Why? It may be a pervasive belief among teachers that word problems are inappropriate for EBs, and EBs may not want to be assigned word problems because they may not understand the problem statement or have anxiety from reading heavy texts. The story of Ms. A was made up, but consider this real story below.
In one of my previous studies, Dr. I worked with preservice mathematics teachers. I recruited promising teacher candidates, and they volunteered to prepare a math lesson and work with one middle school EB in a one-on-one setting. One of the EBs was in an 8th-grade algebra class and the teacher who paired up with this student (let’s call her Ms. B) knew about this fact. The first task Ms. B provided was simple addition and subtraction worksheet, with tasks such as 9+11, which is aligned with 1st-grade mathematics standards according to the Common Core State Standards for Mathematics. The EB, who was in an early algebra class, did all computation tasks with high speed and accuracy and left her teacher bewildered. Later, I asked Ms. B why she prepared those easy tasks for this EB. Ms. B answered, “I knew she was in early algebra and she is capable, but I thought she may have difficulty doing mathematics due to language.”
What do you think about this story? Do you agree with Ms. B? The idea that “EBs cannot do well in mathematics due to language barriers” is a common belief and this belief makes teachers worry about not only word problems but also any mathematical tasks. If a mathematics teacher has this belief, they may think to themselves, “These tasks look too difficult for EBs, so I’ll assign them easier worksheets.” Research has in fact indicated that this belief is not uncommon. In fact, Reeves (2006) found that content teachers, including secondary math teachers, typically believe it is not appropriate to provide word problems to EBs due to their lack of English proficiency. What then can be done about these beliefs?
How to Make Word Problems Accessible to EBs
We would like to suggest rethinking these beliefs as the diagram below suggests. Teachers may observe EBs struggle in understanding word problems because they cannot fully understand the language. However, providing easy tasks or textless problems is not the only solution we can derive from this observation.
From here, we will talk about how we can make a word problem more accessible to EBs.
Generally, there are two directions.
First, we can reduce the language demand using following strategies:
- Identify unnecessary parts and simplify the language
- Provide more blank space on the worksheet, so students don’t feel overwhelmed by heavy dense text, and so they will feel free to jot down their ideas in the space provided.
- Break down the word problem statement into multiple steps.
We need to be careful though. When you reduce the language demand, you must not reduce the mathematical demand. Also, be careful not to reduce the language demand too much because your EBs need to learn language through rich and varied English texts.
Second, we can provide linguistic support:
- Provide mathematically meaningful visuals: You have seen what kind of visuals work better in Chapter 3. Just adding a picture related to the story or context does not help EBs reason mathematically. To learn more about mathematically meaningful visuals for EBs, you can check out the following MTE article and Podcast.
- MTE article: I, J.-Y., & Stanford, J. (2018). Preservice teachers’ mathematical visual implementation for emergent bilinguals. Mathematics Teacher Educator, 7(1), 8–33. https://doi.org/DOI: 10.5951/mathteaceduc.7.1.0008
- Podcast at AMTE site
- Have a set-up time before showing a word problem (I & de Araujo, 2019): During the set-up, a teacher can assess students’ understanding in relation to both mathematics and language to identify what needs to be taught beforehand. In addition, a teacher should assess students’ prior knowledge about the context. For example, when we taught EBs who were from refugee families, we changed a mathematics problem’s context from receiving a birthday gift to receiving an allowance because the EBs did not know their real birth dates and had never celebrated their birthday. If the result of this informal assessments indicates that you need to teach some words, cultural context, and/or prior math concepts, try to reteach or address this before you give the word problem so all students can be on the same page when they are given the word problem.
- Utilize multimedia. The word problem does not always have to be in written form. For example, the story could be delivered through a video as you saw in the 3 Act Task and 5 Act Task introduced in Chapter 3. When you use multimedia that does not include any language, you can include other language activities such as discussion, presentation, or writing a journal to provide rich opportunities for EBs to develop their English skills.
- Allow EBs to use their assets. Let EBs use their most comfortable communicative methods (e.g., written, verbal, drawing) to allow them to ask and respond to each other (if there are proficient bilinguals in the language of your EBs or if you have created a safe environment in your classroom). Most importantly, let EBs use their most comfortable language. Here, the concept of Translanguaging may help.
[The content from here is adapted from Ofelia García’s presentation in 2015]
What do you think translanguaging means? To explain what it means, let’s start by reflecting on the following two short discourses:
Situation 1: Two 5-year-old Emergent Bilinguals who are speakers of Spanish in a bilingual classroom.
T: This tree is bigger. That tree is smaller.
Alicia: [Tries out under her breath]. This tree is grander.
Situation 2: Snack Time for a 5-year-old Emergent Bilingual.
Student A: [Looking out the window and talking to himself] Está lloviendo mucho. [It is raining a lot] Look [telling the others]. It’s washing. There washing afuera [outside]
Question: What did you notice from the EBs’ discourse?
In these dialogues, they used two languages in one sentence or in one situation.
What did you notice from the dialogue above? Did you notice the following?
- EBs are not simply adding English, but using a whole autonomous language with Spanish.
- They are using their own language features in an interrelationship with new ones to make meaning and communicate.
- They are constructing their dynamic bilingual repertoire by adding features to those they already have.
The concept of code-switching, which describes how bilinguals perform discourse as they move from one language to another, is based on the belief that two separate languages exist in the brain of bilinguals. The term, code-switching has often been used negatively, indicating that bilinguals’ use of two languages within a sentence represents imperfect language acquisition.
However, we see the discourse pattern of bilinguals in the examples above as translanguaging, which is a natural way of how bilinguals/multilinguals use their multiple languages with their one language system. We can see that the speakers are actually combining two (or more) languages to communicate as meaningfully as possible. If you find EBs using two languages in one sentence, it does not mean a lack of language acquisition. Rather, they use two languages in an intertwined manner to communicate in the most meaningful way possible at their level. Hence, it is not necessary for teachers (or parents) to push EBs to use only one language in a sentence. If we force them to use only one language, it denies their whole language system and rejects part of who they are and what they can do.
So, which way would you guide your Emergent Bilinguals?
→ Utilize ONLY FEATURES THAT HAVE BEEN APPROVED BY SCHOOL for the task
→ Utilize ANY OF THE FEATURES in their language repertoire to show what they know and can do.
Learners ALWAYS leverage their entire language repertoire in the process of communication. With an understanding of translanguaging, we can change the question, “How do I support students to learn English as a second language?” to “How do I engage students in appropriating the language features associated with English into their own unique language repertoire?”
Translanguaging can support EBs in more than the two ways discussed above. Translanguaging approaches do not reduce language demand but provide a richer and safer language environment so that EBs do not feel overwhelmed by the language demand. By valuing and enacting translanguaging, teachers can value not only students’ culture and language but also their identities as bilinguals (or multilinguals). This section includes how to employ a translanguaging pedagogy for teaching. Translanguaging Pedagogy means the deployment of a speaker’s full linguistic repertoire to learn and develop ways of using language and extend their repertoire and to equalize the positions of learners. For EBs, a Translanguaging Pedagogy removes the handicap of English-only instruction that has been shown to harmful for EBs (Garcia, 2015; Fu, Hadjioannou, & Zhou, 2019).
In the Foreword of the book, The Translanguaging Classroom: Leveraging Student Bilingualism for Learning (García, Johnson, & Seltzer, 2017), Guadalupe Valdés, a researcher in bilingualism at Stanford University, noted that this book is “by far the most compelling example proposed to date of a culturally sustaining pedagogy” (p. vii) with the following explanation.
Too often, proposed pedagogies for cultural responsiveness or relevance have not necessarily invited students to value what they bring or to proudly continue to use features of their full linguistic repertoires in both formal and informal oral and written production for a variety of purposes in and out of school. This book [translanguaging] is different. It explicitly takes the position that past scholarship on language has misunderstood the nature of bilingualism and bilingual practices. It insists that students be invited to foster, maintain, and develop their complex repertoires. It invites teachers to reject static views of Language A versus Language B kept separate and pristine. It urges them to engage thoughtfully and joyfully with the richness of multicompetence in children’s lives.
– Guadalupe Valdés within García, Johnson, & Seltzer (2017, p. vii-viii)
Emphasizing that translanguaging is a pedagogical model grounded on pluralist theory, Fu, Hadjioannou, and Zhou (2019) propose three key tenets for translanguaging practice that can be used in any setting with both emergent and experienced/proficient bilingual students.
- Individuals have a single, unified linguistic repertoire.
- Teachers are co-learners in their classrooms, willing to learn from students, their languages, and their cultures, rather than functioning as the sole possessors of knowledge, “the expert” or the only language instructor in the classroom.
- Translanguaging practice is purposefully and systematically incorporated in both instructional planning and practices.
To help gain a better understanding of how to apply translanguaging pedagogy to a classroom, we will introduce a case of a 5th grade class, which was included in the book, Translanguaging for Emergent Bilinguals (Fu, Hadjioannou, & Zhou, 2019). Mr. Miller is a native English speaker who learned Spanish as a foreign language and the students in the class speak English, Spanish, Polish, Arabic, and Russian. In the classroom, the learning objectives, class schedule, and homework assignments are written in five languages. There is also a word wall of the five languages that Mr. Miller asked students to create, so the students completed the word wall with help from parents and family members.
To prepare for the lesson, Mr. Miller asks the students to do research about school bullying using all of the assets they have including their family members, books and internet resources written in any language and to take notes in the language of student’s choice.
During the lesson, students share their notes in groups in various forms, such as stories, blogs, or real experiences. The students switch back and forth between their home languages and English depending on their audience to make sure everyone understands the conversation.
Next, Mr. Miller reads a storybook on school bullying aloud in English and shows the pictures to the class without stopping the first time. During a second reading, he reads a section and then pauses, letting students talk to their neighbor and discuss what they have learned using any language they choose.
After this, Mr. Miller writes key words related to bullying in English on the board, and invites students to add different home language words under the English words. If they don’t know the word in their home language, they can check the words online or in dictionaries. Phonemic spelling of the words using the Latin alphabet is acceptable.
After checking the key vocabulary words, Mr. Miller asks the students to write individually in any language on their thoughts about the book and any connections they might have between the story and their prior knowledge. After this quick-write, students have a group discussion using translanguaging. When the teacher comes to a group with a language he doesn’t know, the students immediately shift their language to English so the teacher can understand them.
After the group discussion, each group creates a list of questions about bullying they want to investigate and presents them in class. Most questions are written in English, though some questions include words in other languages. Mr. Miller writes the questions on the board including significant non-English words in parentheses. The teacher and students go over the questions together and choose four questions they want to explore, and a new set of student groups discuss the questions.
Before the end of the class, the groups plan how they will find the information to respond to the chosen questions. Mr. Miller closes the class with this statement, “Your resources can be in any language you want, but please share your information tomorrow in English so we can all understand you.”
This is an example of a social studies translanguaging classroom, but you may find a way to apply similar strategies for math classes. Here are some examples of translanguaging strategies for mathematics teachers.
- Give a word problem in English and give students time to discuss and write their solutions in their own languages (or vice versa).
- Let students to see or listen for synonyms and homophones for certain words in English, their home language or both.
- Give EBs an opportunity to demonstrate their knowledge and skills in mathematics in a variety of ways (multiple languages and multiple modes).
- Encourage EBs to use various resources in their home languages including their family members.
You may want to see what a math lesson looks like when it employs a translanguaging pedagogy. You can find one example at the website of CUNY-NYS Initiative on Emergent Bilinguals (CUNY-NYSIEB, ).
Word Problem Analysis Framework
We say teachers need to make mathematical problems accessible to EBs, but what exactly does that mean? According to Luciana de Oliveira (2012), accessible in this statement means providing EBs with “access to the ways in which knowledge is constructed in the content areas—not by simplifying the texts, but by developing teachers’ understanding about how mathematical disciplinary discourse is constructed (p. 195). This section introduces one accessible way to use word problems for EBs by applying the Translanguaging Pedagogy and Word Problem Analysis Framework (Oliveira, 2012).
Here is one example of a word problem:
Moises is saving money to buy a book, which costs $20. He can save $3 per week. How many weeks will it take him to save enough money to buy the book?
This word problem can be divided into three clauses:
Clause 1: Moises is saving money to buy a book, which costs $20.
Clause 2: He can save $3 per week.
Clause 3: How many weeks will it take him to save enough money to buy the book?
de Oliveira (2012) recommends analyzing each “clause” of a mathematical word problem to identify the language demand needed to make the problem accessible for EBs. She proposed five questions based on Huang and Normandia (2008).
- What task is the student asked to perform?
- What relevant information is presented in the word problem?
- What mathematical concepts are presented in the information?
- What mathematical representations and procedures can students use to solve the problem, based on the information presented and the mathematical concepts identified?
- What additional language demands exist in this problem?
The table below is a framework that can be used by teachers or students. Teachers are encouraged to fill out the form in advance, before asking students to analyze the given word problem. We adapted the Framework for Analyzing Word Problems (de Oliveira, 2012) by adding translanguaging components.
|Information Provided in English||Information in a language of choice||Mathematical concept in a language of choice||Mathematical Representation and Procedures|
|Clause 1: Moises is saving money to buy a book, which costs $20.||모세는 $20짜리 책을 사기 위해 저금하고 있다.||The cost of the book: $20|
|Clause 2: He can save $3 per week.||모세는 매주 3달러 모을 수 있다.||Money that Moises can save for 1 week: $3
Per week: 1주
Therefore, $3 can be saved each week.
|Clause 3: How many weeks will it take him to save money to buy the book?||모세가 책을 살 수 있을 때까지 몇 주나 걸릴까?||Money to buy the book ≥ $20
모인 돈이 $20보다 많아야 한다.
The goal of using this framework is to enable teachers to be more proactive in helping EBs learn the ways in which language is used to construct mathematical knowledge. Although this framework shows reading, writing and representation modes, teachers can utilize other language modes, such as speaking or gesturing in classrooms. The teachers should clearly provide directions for using this form so that students can use any language they choose, as this tool is for them to understand the language and mathematical information embedded in a word problem. Sometimes, teachers may give a word problem written in their EBs’ first language and change the second column to translating each clause into English. In this way, EBs can develop not only English but also their first language, and by comparing the two languages, EBs can deepen their understanding of English. After students fill out this form, the teacher can ask students to share their analysis with their group. Students can start solving the problem after the teacher makes sure everyone understands the problem’s language through this framework.
We would like to close this chapter with the following quote from de Oliveira (2012).
Content is never separate from the language through which that content manifests itself. Learning mathematics means learning the language that expresses mathematics. The language demands that this chapter addresses highlight the kind of discipline-specific academic support in language and literacy development that would enable English language learners to be more successful in their mathematics learning (p. 204).
de Oliveira, L. C. (2012). The language demands of word problems for English language learners. In S. Celedon-Pattichis & N. G. Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs (pp. 195–205). National Council of Teachers of Mathematics.
Fu, D., Hadjioannou, X., & Zhou, X. (2019). Translanguaging for emergent bilinguals: Inclusive teaching in the linguistically diverse classroom (First edition). Teachers College Press.
García, O. (2015), The translanguaging current in language education, Presented at åhörarkopior från Symposium 2015. Retrieved from https://www.slideshare.net/NCandrasprak/ofelia-garca-hrarkopior-frn-symposium-2015
García, O., Johnson, S. I., & Seltzer, K. (2017). The translanguaging classroom: Leveraging student bilingualism for learning. Caslon.
Huang, J., & Normandia, B. (2008). Comprehending and solving word problems in mathematics: Beyond key words. In Z. Fang & M. Schleppegrell (Eds.), Reading in secondary content areas: A language-based pedagogy. University of Michigan Press.
I, J. Y., & de Araujo, Z. (2019). An examination of monolingual preservice teachers’ set-up of cognitively demanding mathematics tasks with emergent multilingual students. Research in Mathematics Education, 1–21. https://doi.org/10.1080/14794802.2019.1615980
I, J. Y., & Stanford, J. (2018). Preservice teachers’ mathematical visual implementation for emergent bilinguals. Mathematics Teacher Educator, 7(1), 8–33. https://doi.org/DOI: 10.5951/mathteaceduc.7.1.0008
Reeves, J. (2006). Secondary teacher attitudes toward including English-language learners in mainstream classrooms. The Journal of Educational Research, 99(3), 131–142.