## Teaching Math to English Language Learners: Can Research Help?

#### By Suzanne Irujo, *ELL Outlook™* Contributing Writer

It seems almost unbelievable now, but many people in education used to think that mathematics classes in English would be easy for English language learners (ELLs) because math was less language-dependent than other subjects, as it dealt with numbers. When I was a bilingual teacher in the 1970s, it was routinely recommended that bilingual students be placed in math as their first mainstream subject. People really believed that math was nonverbal.

This belief was so pervasive that somebody had decided that the bilingual classes in my school system could learn math from nonverbal textbooks. These were programmed instruction texts that broke down calculations into their smallest parts and modeled each step for students to copy and then do on their own. They covered nothing but addition, subtraction, multiplication, and division, and most of my third- and fourth-grade students were so bored by the process that they never even made it to division.

After a year of fighting the boredom of nonverbal programmed instruction, I put my students into grade-level math textbooks in English, teaching the lessons orally in Spanish and translating the word problems. Although much better than the previous texts, it was far from ideal. I eventually began teaching one day in Spanish and one day in English so my students would be better prepared for all-English math classes.

Would knowledge of research on teaching math to ELLs have helped me? Probably, but there wasn’t any research on teaching math to ELLs at that time. The first study that I know of wasn’t done until the mid-1980s (Cuevas, 1984). There is still not a lot of research, and there are few areas in which there are enough studies with similar results to be able to draw firm conclusions. But there is enough to provide some guidance for teachers in areas such as language factors that cause difficulty, interactional discourse patterns in the classroom, and the importance of using students’ cultural backgrounds in instruction.

**Language Factors**

Over the years, a respectable body of knowledge has been developed about what kinds of language cause difficulty for ELLs. This research makes it very clear that math is *not* nonverbal. Anybody who has ever tried to teach math to ELLs will have come to the same conclusion, since it is obvious that mathematics involves critical thinking, reasoning, and problem solving.

Most studies of language difficulties in math are descriptive, listing and explaining the kinds of vocabulary that ELLs have trouble with, what sentence structures cause problems, how the correspondence or noncorrespondence of words and symbols affect math learning, characteristics of word problems that make their comprehension difficult, and so forth. These features of the special language of mathematics are often referred to as a *math register.* (In linguistics, a register refers to a variety of a language that is used in a particular situation; in this case, the situation is learning and talking about mathematics.) Here is a summary of what I found when I reviewed this research:

- Vocabulary/semantics difficulties:
- Words with math meanings that are different from their everyday meanings (e.g.,
*set, point, field, column, sum, random, table, altogether, round, equals*) - Words or phrases that are “conceptually dense,” in that they convey very complex meanings (e.g.,
*exponent, coefficient*) or combine two or more concepts to form a new concept (e.g.,*common denominator, least common multiple*) - Multiple terms for the same thing (
*add, sum, plus, combine, put together, increased by; subtract, decreased by, take away, minus, less, difference; multiply, times, product; divide, into, quotient*)

- Words with math meanings that are different from their everyday meanings (e.g.,
- Vocabulary/syntax difficulties (understanding a concept is harder when the concept is made up of the relationship between two words):
- All numbers
*greater/less*than X - Mary earns
*5 times as much as*John - Joseph is
*as*old as Mary - Mary is 6 years
*older*than John *Twenty*(used as noun) is five times X- When 10
*is added*(passive) to X *Two numbers, the sum of which*is 1*Two numbers, whose product is*1,*are reciprocals of each other**By what percent is*16*increased to make*24?*Divided by*versus*divided into*- Lack of one-to-one corre
__spon__dence between symbols and words:*8 divided by 2*is not 8)2*The number a is 5 less than the number b*is not:*a*= 5 –*b*(it’s a =*b – 5*)

- All numbers
- Discourse difficulties:
- Logical connectors (
*if… then, if and only if, because, that is, for example, such that, but, consequently, either… or*)- These may signal similarity, contradiction, cause/effect, reason/result, chronological sequence, or logical sequence

- References of variables (variables are the
*number*of things,*not*the things themselves)*There are 5 times as many students as teachers in the math department*(the correct equation is 5*t*=*s*, not 5*s*=*t*)*Three times a number is 2 more than 2 times the number*(*number*refers to the same number both times)*If the first number is 2 times the other, find the number*(what do*first number, the other,*and*the number*refer to?)

- Cultural difficulties (notation of division problems; use of periods and commas; units of measurement; use of fractions; application of rules versus analysis and problem-solving)

- Logical connectors (

Recently, the studies of language difficulties in math have become more explanatory. They go beyond merely lists of the difficulties to an explanation of why they occur. For example, Ron (1999) develops the concept of “mathematized language,” which can help explain why many ELLs progress well with the language of mathematics at first but then hit a plateau. This theory begins with everyday language, which is acquired naturally through social interaction. Mathematized language is similar to everyday language, but makes the mathematical concepts that are present in the everyday language explicit. Mathematized language can be used to help build up mathematical language. Ron provides an example of how a child uses everyday language to talk about wanting to buy a doll but not having enough money. Through natural acquisition with some instruction, the child learns to state this in mathematized language, by saying how much money she has and how much the doll costs and asking how much more money she needs. This mathematized language makes the transition to the language of mathematics easier. The language of mathematics then allows the child to verbalize the fact that she has to add some unknown amount to the money she has ($15) so it equals the price of the doll ($22), and finally move to symbolic language: 15 + *x* = 22. At each step in this process, the language of math must be more consciously taught and learned; if this does not happen, children will have difficulty reaching the stage of symbolic language.

Another area of language difficulties that has received a good deal of research attention is word problems. In order to solve a word problem, ELLs must be able to understand the language in the problem, interpret that language so they can identify the math relations and understand what the problem is asking, and convert the language and the math relations to abstract symbols. All of this is made more difficult by the fact that word problems are artificial situations described using the mathematical language of problem solving, which makes it difficult to use reading skills learned in other contexts to help understand the problem.

This subregister of the register of mathematics language has been called “word-problemese.” Here are some of the difficulties it creates:

- The language used in the problem is often more complex than it needs to be order to do the math.
- The language lacks redundancy, so there are no repetitions or expansions, both of which help learners construct and corroborate meaning.
- Word problems are either set in artificial contexts or lack context, which can create confusion.
- If students don’t fully understand the language that is used to describe the situation in a word problem, they will have difficulty connecting the mathematical operations to the situation.
- Illustrations accompanying word problems often don’t aid in the comprehension of the problem.

Students who have difficulty understanding word problems often adopt strategies for solving them that may or may not work. They tend to pay more attention to the mathematical content of the problem than to the verbal content. They look for key words (*total* or *all together* means addition;*how many more* or *how many left* means subtraction), or guess what operation to use based on the relative size of the numbers in the problem. Instruction in solving word problems that breaks the process down into specific steps and teaches comprehension strategies that are specific to math problems could help ELLs understand the problems and thus be better able to solve them.

Another thing that can help is to find out what strategies ELLs use when they solve word problems and to design instruction to build on successful strategies and eliminate unsuccessful ones. A good first step toward this goal is a study done by Celedón-Pattichis (2003), who used think-alouds to find out how ELLs in grades 6 to 8 approached word problems. They used successful strategies such as reading the problem twice, translating the problem into Spanish, inferring meaning, using symbols to help understand the math, and ignoring irrelevant words. Problems arose through misinterpreting words that students incorrectly assumed were homophones (such as *many* and*money*, or *than* and *then*), and misinterpreting math symbols (such as reading 3 1/2 as “thirty-one slash two”). They were also confused by math language being used in a non-math context (for example, a reference to a “2 1/2 can” in a problem about food containers of different sizes).

**Classroom Discourse**

Looking at how teachers and students use mathematical language in classrooms and how they organize instruction is another area of research that can be helpful to teachers of ELLs. There has been more interest in classroom discourse since the math reform movement of the 1990s, because part of that reform movement included putting a much greater emphasis on ensuring that students are able to explain their reasoning, their use of strategies, and their solutions. Researchers are exploring questions such as whether and how collaborative learning facilitates math learning, whether and how teachers and students use mathematical language in classrooms, and how teachers’ beliefs affect the way they organize their classrooms.

One example of this is the work being done on collaborative learning. It has long been assumed that ELLs benefit from small-group interaction, but nobody has given much attention to questions about factors that make group interaction more or less effective. In an interesting comparison of two algebra classes implementing a new curriculum that emphasized communication in groups, Brenner (1998) found an interesting contrast. In a sheltered algebra class made up of all ELLs taught by a teacher who consciously implemented sheltered instruction strategies, the adjustments made by the teacher to accommodate the students’ lack of proficiency in English undermined the program’s cooperative group structure. The result? Students did not develop mathematical communication skills. They did not participate in large group discussions, and the teacher couldn’t understand what they were doing in small-group activities. The teacher was thus deprived of the feedback she needed in order to assess what students understood. When they asked for help, she did not have the information she needed to be able to guide them in their own pursuit of algebraic understanding. Even though the teacher continued to implement the small-group structure of the program, she felt she had to rely heavily on transmitting knowledge. Students were reluctant to rely on each other, and thus did not take advantage of the safe environment of the small group to learn how to express themselves mathematically. In contrast, the teacher of a mainstream algebra class with a substantial minority of ELLs used the small-group structure of the program to give ELLs more opportunities to participate. This gave them enough confidence to begin to take part in whole-class discussions. These two teachers implemented the same program. In one class, there was very little math communication, and it was mostly oriented toward simple answers and procedural descriptions. In the other class, there was extensive math communication in both small groups and in whole-class discussions.

Another way in which qualitative studies are valuable is that they can document what actually goes on in classrooms. What kind of language do teachers use when they teach math to ELLs? Although qualitative research can’t provide data that is generalizable, it can give us glimpses into real classrooms. Khisty (1993) has done this with her study of bilingual teachers who were deemed to be effective at teaching fifth-grade math. Surprisingly, the observations showed that these teachers used very few words or phrases from the register of mathematics. There was a lot of interaction in their classes, but the language of math was absent. For example, in teaching about fractions, teachers identified the *denominator* as the “number on the bottom,” without ever explaining its mathematical meaning. They explained how to subtract 1/3 from 2/3 by saying “two minus one is one,” without identifying numerators or denominators. Khisty summarized her findings by saying that very little math language was used, and when it was, it was often incorrect, inappropriate, ambiguous, or unrelated to the development of meaning.

It may be that these teachers were consciously trying to simplify their language to make it more comprehensible to their students. (In spite of being bilingual teachers, they made little use of Spanish.) Or it may be that they were unaware of the importance of using and explicitly teaching the language of math. Either way, this study points to a need for improved mathematics teacher education.

**Sociocultural Issues**

The idea of using students’ culture to make what they are learning more relevant (and thus more likely to be learned) has been around since the first federal funding of bilingual programs, when funding of grant proposals depended on grantees’ showing that they would use students’ language*and* culture. “Culture” was often seen in a very superficial way, however, at the level of “ferias and folktales.”

Culture is now defined at a much deeper level, with emphasis on beliefs and values, which are often deeply rooted in a group’s history and traditions. Reyes and Fletcher (2003) draw on these traditions to incorporate the contributions to mathematics made by a particular culture into the math instruction used with members of that culture. They give examples of using the Mayan calendar and Mayan mathematics in math programs for migrant students in the Rio Grande Valley of Texas. This mathematical culturally relevant pedagogy is referred to as “ethnomathematics.”

A very different perspective on integrating culture in mathematics teaching is taken by Norma González and her colleagues in Project Bridge (Linking Home and School: A Bridge to the Many Faces of Mathematics). This is an outgrowth of the well-known Funds of Knowledge for Teaching, developed by Luis Moll and his colleagues at the University of Arizona. Funds of Knowledge for Teaching trains teachers in ethnography so they can learn about the knowledge and learning practices of their students’ families and communities, and integrate this into their teaching. Researchers and teachers working within this framework learn about home and community practices, and develop teaching materials that use this knowledge. Project Bridge focused the Funds of Knowledge framework on community practices that could be used in teaching mathematics. Examples (in various chapters of McIntyre, Rosebery, & González, 2001) include materials derived from community agricultural practices, from weaving and gardening, and from students’ knowledge of the construction industry. The focus on what households actually do allows teachers to incorporate the dynamic nature of culture into teaching and learning within the context of a specific community.

**Conclusion**

So what can research tell us about what individual teachers should be doing to improve the math achievement of ELLs? With the caveat that these recommendations are not based on a complete review of the literature, I think that the studies I’ve looked at here have the following implications:

- All teachers of mathematics to ELLs (actually, to any student) must understand the math they are teaching, and must know the academic language associated with it, well enough to be able to use it automatically in their classroom interactions with students. They must also understand the importance of explicitly teaching the language of mathematics, and know how to effectively teach academic language to ELLs.
- All teachers of mathematics to ELLs must know how to structure small-group interaction so students can and will take advantage of the safe environment of small groups to use academic math language as they talk about math processes and concepts. They must know how to scaffold students’ interactions in small groups and how to gradually reduce the scaffolding so students will become comfortable interacting in whole-class situations.
- All teachers of mathematics to ELLs must become familiar with their students’ backgrounds in order to make the math curriculum culturally relevant by drawing on the knowledge and resources of students’ homes and communities.

In the end, it comes down to individual teachers, as it always does. Teachers must know what they need to do to be effective, know how to do it, and have the support they need to do it. Future articles will look more closely at the academic language of math and how teachers can address the particular difficulties ELLs have with the language of math, and at teaching strategies that can help improve the effectiveness of math teaching for ELLs.

**References**

Brenner, M. E. (1998). Development of mathematical communication in problem solving groups by

*Bilingual Research Journal 22,*214-244.

Celedón-Pattichis, S. (2003). Constructing meaning: Think-aloud protocols of ELLs on English

*Educators for Urban Minorities, 2*(2), 74-90.

Cuevas, G. (1984). Mathematics learning in English as a second language. *Journal for Research*

*in Mathematics Education, 15,*134-144.

Khisty, L. L. (1993). A naturalistic look at language factors in mathematics teaching in bilingual

*Proceedings of the third National Research Symposium on Limited English Proficient Student Issues: Focus on Middle and High School Issues*. Washington, DC: U.D. Department of Education, Office of Bilingual and Minority Language Affairs. Available: http://www.ncela.gwu.edu/pubs/symposia/third/khisty.htm

McIntyre, E., Rosebery, A., & Gonález, N. (2001). *Classroom diversity: Connecting curriculum*

*to students’ lives.*Portsmouth, NH: Heinemann.

Reyes, P., & Fletcher, C. (2003). Successful migrant students: The case of mathematics. *Journal*

*of Curriculum and Supervision 18,*306-333.

Ron, P. (1999). Spanish-English language issues in the mathematics classroom. In W. Secada,

*Changing the faces of mathematics: Perspectives on Latinos*(pp. 22-33). Reston, VA: National Council of Teachers of Mathematics.