English learners (ELs) are currently about five million students in the U.S.* If we want these students to have access to equitable math instruction*, *we need to first move past deficit views of those learners and contradict common sense notions of what they need*. Research suggests that EL instruction that supports student achievement has two characteristics: a view of language as a resource, not a deficiency; and an emphasis on academic achievement, not only learning English (Gándara & Contreras, 2009). Overall, research shows that students from non-dominant communities need access to curricula, instruction, and teachers effective in supporting academic success for this student population. General characteristics of such environments are that instruction provide “abundant and diverse opportunities for speaking, listening, reading, and writing” and “encourage students to take risks, construct meaning, and seek reinterpretations of knowledge within compatible social contexts” (Garcia & Gonzalez, 1995, p. 424). Teachers with documented success with students from non-dominant communities share some characteristics: a) high commitment to students’ academic success, b) high expectations for all students, and c) a rejection of models of their students as intellectually disadvantaged[1].

Research specific to mathematics (Moschkovich, 2012, 2013a, 2013b) suggests that mathematics instruction for ELs should:

- Support English learners’ participation in mathematical discussions as they learn English (Moschkovich 1999, 2002, 2007a, 2007b, 2007c).
- Focus on mathematical practices such as reasoning and justifying, not accuracy in using individual words, and address much more than vocabulary.
- Treat home and everyday language as resources, not deficits. Draw on multiple resources available in classrooms—objects, drawings, graphs, gestures—as well as home languages and experiences outside of school (Moschkovich, 2000, 2014a, 2014b).

These recommendations are based on research that runs counter to commonsense notions of what it means to learn math and deficit views of students who are learning English.

**Contradicting deficit views of bilingual mathematics learners**

Over my 25 years doing research that focuses on the resources that emergent bilinguals bring to mathematical discussions, I have heard multiple expressions of deficit views of bilingual mathematics learners. These can appear in many guises: how we frame English learners, i.e., as “a problem,” facing more obstacles than monolingual learners, having deficits that prevent them from participating in mathematical discussions, or needing to learn English before they can have access to grade-level math instruction. Deficit views typically focus only on obstacles these students face and neglect to consider the resources these students bring for reasoning mathematically, and can go as far as claiming that bilingual learners’ language practices (i.e., using their first language) are a deficit or evidence of an imagined cognitive deficit. Research findings contradict these views. Overall, there is strong evidence suggesting that bilingualism does not impact mathematical reasoning or problem solving (Bialystok, 2001). There are also relevant findings regarding two common practices among bilingual/multilingual mathematics learners, during arithmetic computation or mathematical discussions (for a review, see Moschkovich 2007b, 2009).

*Using two languages during computation: *Older bilingual students who learned arithmetic facts in their first language may carry out arithmetic computations in a preferred language, usually the language in which they learned arithmetic. However, reported differences in calculation times between adult monolinguals and bilinguals are miniscule. There is evidence suggesting that switching languages for arithmetic computation does not affect the quality of conceptual reasoning. Language switching can be swift, highly automatic, and facilitate rather than inhibit solving word problems in the language of instruction, provided the student’s proficiency in the language of instruction is sufficient for understanding the text of a word problem. These findings suggest that classroom instruction should allow bilingual students to choose the language they prefer for arithmetic computation and support all students in learning to read and understand the text of word problems in the language of instruction.

*Using two languages during mathematical discussions: *Another common practice among emergent bilinguals is switching languages during a sentence or conversation, what linguists call “code-switching” or “trans-languaging.” Research does not support a view of code switching as a deficit itself or as a sign of deficiency in mathematical reasoning. Researchers in linguistics agree that code-switching is not random or a reflection of language deficiency—forgetting a word or not knowing a concept. It is crucial to avoid deficit conclusions regarding code switching and mathematical thinking. We should not conclude that bilingual students switch into their first language because they do not remember a word, are missing English vocabulary, or do not understand a mathematical concept (Moschkovich, 2007b). 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. Bilingual speakers have been documented using their two languages as resources for mathematical discussions, for example first giving an explanation in one language and then switching to the second language to repeat the explanation (Moschkovich, 2007a).

Lastly, deficit views of bilingual math learners sometimes appear as beliefs that some languages are just better than others for doing mathematics. Barwell (2009) made two observations relevant to that belief: 1) “all languages are equally capable of developing mathematics registers, although there is variation in the extent to which this has happened,” and 2) “the mathematics registers of different languages . . . stress different mathematical meanings.” These differences in mathematics registers, however, should not be construed as a reflection of differences in learner’s abilities to reason mathematically or to express mathematical ideas. Nor should we assume that there is a hierarchical relationship among languages that have different ways available to express academic mathematical ideas.

Deficit views of these learners restrict access to the resources they need for learning mathematics. Students need to have the opportunity to use the full range of their linguistic and non-linguistic meaning-making resources during mathematics because these resources are useful for communicating and learning math with understanding. When school policies or practices limit a child from the full repertoire of his/her linguistic and /or cultural resources for learning, then the child is also restricted from learning math with understanding. Problems may arise for emergent bilingual students in the mathematics classroom because of the lack of access to crucial meaning-making resources rather than limited ability, limited language proficiency, or cultural differences.

**Equitable teaching practices for ELs in mathematics classrooms**

Changing deficit views of bilingual math learner is the first step in equitable math teaching, but it is not sufficient. The next step is contradicting common views of learning math as memorizing and, instead, develop teaching practices that focus on supporting students understanding of math ideas. In many instances, ELs can get stuck in a cycle of remedial courses that focus on memorizing and have limited access to instruction that focuses on understanding (NASEM, 2019).

Students who are learning English need to develop the full range of math proficiency that includes BOTH procedural fluency and conceptual understanding. They need to have opportunities to engage in mathematical communication, because this is a central way to develop understanding. Now, one might think “Sure, classroom discussions may support conceptual understanding. But ELs can’t participate in mathematical discussions because they are just learning English.”

That may seem like a common-sense claim. However, research shows that ELs, even as they are learning English, can participate in discussions where they grapple with important mathematical content[2]. Instruction for this population should not emphasize low-level language skills over opportunities to actively communicate about math ideas. One of the goals of math instruction for students who are learning English should be to support all students, regardless of their proficiency in English, in participating in discussions that focus on important math ideas, rather than on pronunciation, vocabulary, low-level linguistic skills, or arithmetic computation. By learning to recognize how ELs express their math ideas as they are learning English, teachers can maintain a focus on the math concepts as well as on language.

**How can teachers support ELs in learning mathematics with understanding?**

*There are several ways teachers can support students who are learning English in learning math with understanding:*

- Focus on students’ mathematical reasoning, not accuracy using language.
- Support students’ conceptual understanding.
- View both every-day and home languages as resources for communicating mathematically.
- Use high quality resources to focus on student conceptual understanding (not only procedural fluency), and support productive whole class math discussions.

*Focus on students’ mathematical reasoning, not accuracy in using language.* Instruction should focus on uncovering, hearing, and supporting students’ mathematical reasoning, not on accuracy in using language (either English or a student’s first language). When the goal is to teach for understanding and engage students in mathematical practices (CCSS, 2010), student contributions may first appear in imperfect language. Teachers should not be sidetracked by expressions of mathematical ideas expressed in imperfect language. Instead, teachers should first focus on promoting meaning, no matter what kind of language students may use. Eventually, after students have had ample time to engage in mathematical practices both orally and in writing, instruction can then carefully consider how to move students toward increasing accuracy in using language to express their ideas.

As a teacher, it can be difficult to understand the mathematical ideas in students’ talk in the moment. However, it is possible to take time after a discussion to reflect on the math content of student contributions and design subsequent lessons to address these mathematical ideas. But, it is only possible to uncover the mathematical ideas in what students say if students have the opportunity to participate in a discussion and if this discussion is focused on mathematics. Understanding and re-phrasing student contributions can be a challenge, perhaps especially when working with students who are learning English. It may not be easy (or even possible) to sort out what aspects of what a student says are due to the student’s conceptual understanding or English language proficiency. However, if the goal is to support student participation in a mathematical discussion and in mathematical practices, determining the origin of an error is not as important as listening to the students and uncovering the math content in what they say and do. In order to focus on the mathematical meanings that learners construct rather than the mistakes they make or the obstacles they face, teachers need to recognize the emerging mathematical reasoning that learners are constructing in, through, and with emerging language (and as they learn to use multiple representations).

*Support students’ conceptual understanding and provide students access to more than computation. *A research-based definition of mathematical proficiency from the National Research Council (Kilpatrick, Swafford, & Findell, 2001) describes five intertwined strands: I address only two components, procedural fluency and conceptual understanding. Fluency in performing mathematical procedures is what most people imagine we mean when we say “learning mathematics.” Conceptual understanding is more difficult to define. It involves connections, reasoning, and meaning that learners (not teachers) construct. Conceptual understanding is more than performing a calculation accurately and quickly, or memorizing a definition or theorem. It involves understanding why a result is the correct answer or what a result means (i.e., explaining or showing with a picture why the result of multiplying 1/2 by 2/3 is smaller than ½). Conceptual understanding involves connecting representations (such as words, drawings, symbols, diagrams, tables, graphs, equation, etc.), procedures, and concepts (Hiebert & Carpenter, 1992). If a student understands the procedure for generating equivalent fractions, we would say they have made connections between two representations, the number symbols (1/2, 4/8) and an area model, and can show how to transform 1/2 into 4/8 using a rectangle first divided into two equal parts and later divided into eight equal parts.

*Why is conceptual understanding important? *One might think, “OK fine, so researchers think students need to draw pictures and explain a procedure, but what is the big deal about conceptual understanding? Why can’t students just learn their multiplication facts or learn that the right procedure to create equivalent fractions is to multiply the top and bottom by the same number and be done with it? I certainly don’t think I understand most of the arithmetic I learned and yet I have made it through school. Why does a child’s learning need to include conceptual understanding?” One answer is that conceptual understanding and procedural fluency are related, even if we, as adults, do not now remember understanding a procedure when we learned it. Research (Bransford, Brown, & Cocking, 2000) has shown that people remember better, longer, and in more detail if they understand, actively organize what they are learning, connect new knowledge to prior knowledge, and elaborate. Thus, children will remember procedures better, longer, and in more detail if they actively make sense of procedures and connect procedures to concepts and representations. Rehearsal (repeating something over and over) may work for memorizing a grocery list (and, even then, organizing the list will improve memorization). Rehearsal, however, is not the most efficient strategy for remembering how to perform demanding cognitive tasks such as arithmetic operations. The best way to remember is to understand, elaborate, and organize what you know.

*Why is communication important for learning mathematics? *One might think, “OK, I can see why children need to develop conceptual understanding, but what is all the fuss about communication in the math classroom? I always did math by myself at my desk.” Communication is important because it supports conceptual understanding. The more opportunities a learner has to make connections among multiple representations, the more opportunities that learner has to develop conceptual understanding. But not all kinds of communication will support conceptual understanding in mathematics. Communication needs to be focused on important math ideas. Classroom communication that engages students in evidence-based arguments by focusing on explanations and justifications builds conceptual understanding. Communication should include multiple modes (talking, listening, writing, drawing, etc.), because making connections among multiple ways of representing math concepts is central to developing conceptual understanding.

*How can teachers plan lessons for ELs that balance attention to conceptual understanding, mathematical practices, and the language demands of talking to learn mathematics? *Teachers can support EL student talk during whole class discussions by using a variety of teacher talk moves, including a variety of participation structures beyond whole class discussions. Resources are available (many free and online) for designing math lessons that pay attention to language. Teachers can work in teams (math, language, and English as a Second Language specialists) to select and use lessons that pay attention to both language and important math. Even when remedial work is necessary (i.e., to facilitate efficient recall of math facts), teachers can embed practicing procedures within lessons that include important math concepts and support student talk. **Repetitive worksheets and rehearsal of procedures are not the optimal ways to remediate missing procedural skills.** Translation and cognates can support students who have had instruction in mathematics in their first language. Lesson design needs to draw on exemplary practices and lessons based on current research to design lessons that pay attention to both conceptual understanding and student talk: approaches to mathematical discussions (Smith & Stein, 2011) and teacher talk moves (Chapin et al, 2003, 2009); the Framework for English Language Proficiency Development Standards[3]; and open source sample student math activities that pay attention to language[4].

**References**

American Educational Research Association (2006). Do the math: Cognitive demand makes a difference. Research Points, 4(2), 1-4.

Barwell, R. (Ed.) (2009). Multilingualism in mathematics classrooms: Global perspectives. Bristol, UK: Multilingual Matters.

Bransford, J., Brown, A., Cocking, R. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.

Bialystok, E. (2001). Bilingualism in Development: Language, Literacy, and Cognition. New York: Cambridge University Press, 2001.

Chapin, S., O’Connor, C., & Anderson, N. (2003). Classroom Discussions: Using Math Talk to Help Students Learn, Grades 1-6, Sausalito, CA: Math Solutions Publications.

Chapin, S., O’Connor, C., & Anderson, N. (2009). Classroom Discussions: Using Math Talk to Help Students Learn, Grades K-6. Math Solutions.

Common Core State Standards. (2010). Common Core State Standards for Mathematical Practice. Retrieved from http://www.corestandards.org/Math/Practice.

Gándara, P., & Contreras, F. (2009). The Latino education crisis: The consequences of failed social policies. Cambridge, MA: Harvard, University Press.

Hiebert, J. and Carpenter, T. (1992). Learning and teaching with understanding. In D. Grouws (Ed.), Handbook of Research in Mathematics Teaching and Learning. New York: Macmillan, pp. 65-97.

Khisty, L. L. (1995). Making inequality: Issues of language and meanings in mathematics teaching with Hispanic students. In W. G. Secada, E. Fennema, & L. B. Adajian (Eds.), New directions for equity in mathematics education (pp. 279-297). New York: Cambridge University Press.

Kilpatrick, Swafford, & Findell (2001). Adding it up: helping children learn mathematics. National Research Council. Washington, DC: National Academy Press. **https://alearningplace.com.au/wp-content/uploads/2016/09/Adding-It-Up_NAP.pdf**

Moschkovich, J. N. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19(1), 11-19.

Moschkovich, J. N. (2002). A situated and sociocultural perspective on bilingual mathematics learners. *Mathematical Thinking and Learning*, N. Nassir and P. Cobb (Eds.), 4(2&3), 189-212.

Moschkovich, J. N. (2007a). Bilingual mathematics learners: How views of language, bilingual learners, and mathematical communication impact instruction. In N. Suad Nasir and P. Cobb (Eds.), *Diversity, equity, and access to mathematical ideas* (pp. 89-104). New York, NY: Teachers College Press.

Moschkovich, J. N. (2007b). Using two languages while learning mathematics, *Educational Studies in Mathematics*,* 64*(2), 121-144.

Moschkovich, J. N. (2007c). Examining mathematical discourse practices. *For the Learning of Mathematics*, *27*(1), 24-30.

Moschkovich, J. N. (2009). Using two languages when learning mathematics: How can research help us understand mathematics learners who use two languages? Research Brief and Clip, National Council of Teachers of Mathematics, available online at http://www.nctm.org/uploadedFiles/Research_News_and_Advocacy/Research/Clips_and_Briefs/Research_brief_12_Using_2.pdf

Moschkovich, J.N. (2012). Mathematics, the Common Core, and language: Recommendations for mathematics instruction for English Learners aligned with the Common Core. Commissioned Papers on Language and Literacy Issues in the Common Core State Standards and Next Generation Science Standards, pp. 17-31. Proceedings of “Understanding Language” Conference. Palo Alto, CA: Stanford University.

Moschkovich, J. N. (2013a). Equitable practices in mathematics classrooms: Research based recommendations. *Teaching for Excellence and Equity in Mathematics*, 5, 26-34.

Moschkovich, J. N. (2013b). Principles and guidelines for equitable mathematics teaching practices and materials for English language learners. *Journal of Urban Mathematics Education, *July 2013, Vol. 6, No. 1, pp. 45–57*.*

Moschkovich, J. N. (2014a). Building on student language resources during classroom discussions. In M. Civil & E. Turner (Eds.) *The Common Core State Standards in mathematics for English language learners: Grades K-8*. Alexandria, VA: TESOL International Association, 7-19.

Moschkovich, J. N. (2014b). Language resources for communicating mathematically: treating home and everyday language as resources. In T. Bartell & A. Flores (Eds.), *Embracing Resources of Children, Families, Communities and Cultures in Mathematics Learning*. TODOS Research Monograph, Volume 3. San Bernadino, CA: Create Space Independent Publishing Platform, 1-12.

National Academies of Science, Engineering, and Medicine (2019). English Learners in STEM subjects: Transforming classrooms, schools, and lives. D. Francis (Ed.). NRC: Washington, DC.

Secada, W. (1991). Degree of bilingualism and arithmetic problem solving in Hispanic first graders. *Elementary School Journal*, *92*(2), 213-231.

Smith, M. S., & Stein, M. K. (2011). Five practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics.

[1] Curriculum policies for ELs in mathematics should follow the guidelines for traditionally underserved students (AERA, 2006), such as instituting systems that broaden course-taking options and avoiding systems of tracking students that limit their opportunities to learn and delay their exposure to college-preparatory mathematics coursework.

[2] For examples of lessons where ELs participate in mathematical discussions, see Moschkovich, 1999, 2007a, and 2012.

[3] The ELPD framework was developed by the Council of Chief State School Officers, the English Language Proficiency Development Framework Committee, in collaboration with the Council of Great City Schools, the Understanding Language Initiative at Stanford University, and World-Class Instructional Design and Assessment, with funding support from the Carnegie Corporation of New York. Full document is available online at http://www.ccsso.org/Documents/2012/ELPD%20Framework%20Booklet-Final%20for%20web.pdf

[4] The document “Language of Math Task Templates” is available on the Understanding language web site

“Reading and Understanding a Math Problem,” a task for supporting academic literacy with word problems, is on pages 37-40 in the document. “Speaking Mathematically,” a task for supporting academic literacy with word problems, is on pages 47-49. The purpose of the task “Reading and Understanding a Word Problem” is to support students in learning to approach a mathematics problem by giving students tools for learning to read, understand, and extract relevant information from a problem. The task “Mathematically Speaking[4]” gives students the opportunity to solve a problem and then explain and discuss how they arrived at their solution using targeted vocabulary.

**Author Biography**

**I write as an advocate for an equitable and high quality mathematics education for all students, in particular Latinx students as well as students from other non-dominant populations. My remarks represent my own professional experience as a mathematics instructor at the university level and as a researcher in mathematics education for over 25 years. My career in mathematics education began when, after receiving a B.S. in Physics, I taught Algebra courses as a lecturer in the Mathematics Department at San Francisco State University. I received my Ph.D. in mathematics education in 1992 and have been conducting research in classrooms since then. I have been involved in mathematics education at many levels: I have served as a member of the editorial panel for the Journal for Research in Mathematics Education and as the Chair of the AERA Special Interest Group for Research in mathematics Education (2004-2006). I am the author of many research articles and chapters in edited books. I teach a course that introduces future secondary mathematics teachers to evidence-based research in mathematics education and courses for Ph.D. students in mathematics education. My research for the past 30 years has focused on the study of the relationship between language and learning mathematics, especially for Latinx students who are learning English. I am originally from Argentina and my first language is Spanish.**