DOCUMENT RESUME SE 065 155 ED 456 042 Celedon-Pattichis, Sylvia AUTHOR Constructing Meaning: Think-Aloud Protocols of ELLs on TITLE English and Spanish Word Problems. 1999-04-00 PUB DATE 34p.; Paper presented at the Annual Meeting of the American NOTE Educational Research Association (Montreal, Canada, April 19-23, 1999). Speeches/Meeting Papers (150) Research (143) Reports PUB TYPE MF01/PCO2 Plus Postage. EDRS PRICE Cultural Influences; *English (Second Language); *Language DESCRIPTORS Processing; Linguistics; *Mathematics Education; Mexicans; Middle Schools; Social Influences; *Word Problems (Mathematics) ABSTRACT This one-year qualitative study analyzed how nine middle school English language learners (ELLs) of Mexican descent constructed meaning on think-aloud protocols of Spanish and English word problems. Strategies used by these students to process information from English to their native language included translating to Spanish, reading the problem at least twice, inferring meaning, understanding mathematical symbols, and ignoring words that were irrelevant to the solution. The students failed to construct meaning in both languages when the word problems mixed mathematical with natural language. The findings indicate the importance of using students' sociocultural and linguistic experiences to make mathematical connections between natural language and the language that is specific to (Author/ASK) mathematics. (Contains 39 references.) Reproductions supplied by EDRS are the best that can be made from the original document. Constructing Meaning: ELLs and Mathematics 1 Running Head: CONSTRUCTING MEANING: ELLS AND MATHEMATICS SCOPE OF INTEREST NOTICE The ERIC Facility has assigned this document for processing to: In our judgment, this document is also of interest to the Clear: inghouses noted to the right. Indexing should reflect their special points of view. Constructing Meaning: Think-Aloud Protocols of ELLs on English and Spanish Word Problems Sylvia Celedón-Pattichis University of New Mexico Paper presented at the Annual Meeting of the American Educational Research Association, Montréal, Canada, April 19-23, 1999. U.S. DEPARTMENT OF EDUCATION ffice of Educational Research and Improvement EDUCATIONAL RESOURCES INFORMATION PERMISSION TO REPRODUCE AND CENTER (ERIC) DISSEMINATE THIS MATERIAL HAS This document has been reproduced as BEEN GRANTED BY 'ceeived from the person or organization originating it. 0 Minor changes have been made to improve reproduction quality. -KIL=6(.i Points of view or opinions stated in this document do not necessarily represent TO THE EDUCATIONAL RESOURCES official OERI position or policy. INFORMATION CENTER (ERIC) 2 BEST COPY AVAILABLE Constructing Meaning: ELLs and Mathematics 2 Abstract This one-year qualitative study analyzed how nine middle school English language learners (ELLs) of Mexican descent constructed meaning on think-aloud protocols of Spanish and English word problems. Strategies used by these students to Spanish, process information from English to their native language included translating to reading the problem at least twice, inferring meaning, understanding mathematical symbols, and ignoring words that were irrelevant to the solution. The students failed to construct meaning in both languages when the word problems mixed mathematical with natural language. The findings indicate the importance of using students' sociocultural and linguistic experiences to make mathematical connections between natural language and the language that is specific to mathematics. 3 Constructing Meaning: ELLs and Mathematics 3 Constructing Meaning: Think-Aloud Protocols of ELLs on English and Spanish Word Problems There is a paucity of studies that specifically address issues regarding the learning of mathematics by English language learners (ELLs)', especially at the middle to secondary level. Literature suggests that first and second language factors affect mathematics learning (Aiken, 1971; Cuevas, 1984; Mestre, 1988; Secada, 1991; Khisty, 1995; Ron, 1999). According to Ron (1999), there are two misconceptions about the role of language in mathematics. First, the language of mathematics is considered to be as easily acquired as the natural language, or the everyday language. Second, if a person is bilingual, then the assumption is that "...he or she automatically knows the language of mathematics in both languages" (p. 23). While it is true that some mathematical symbols has no linguistic are shared in many languages, this does not imply that mathematics concepts. Given the increase in the number of Spanish-speaking students (U.S. Department of Education, 1994) and the current emphasis on learning through word problems (NCTM, 1989, 2000), questions remain on how ELLs use language to negotiate meaning. Even though researchers have recognized the important role that language plays in mathematics performance (Aiken, 1971), they have not always acknowledged its equally crucial role in the process of acquiring mathematical concepts and skills in a second language. To better understand the successes and challenges ELLs encounter as they solve word problems during the initial stages of second language acquisition, this study addresses the following questions: 1) How do ELLs use language to negotiate mathematical meaning given five English and five Spanish word problems? 2) What 4 Constructing Meaning: ELLs and Mathematics 4 problem-solving strategies (i.e., translation, reading the problem at least twice, inferring, etc.) do ELLs use? To address these questions, I present specific examples from word problems in which language interacts with problem-solving performance among nine students of Mexican descent. First, theories in learning a second language, including implications for mathematics, are discussed. Second, the mathematical problem-solving process is presented. Next, I describe the mathematics register to address difficulties language can create for ELLs when solving word problems. A detailed analysis of the findings follows from the English and Spanish think-aloud protocols to discuss how particular language used in word problems helped (or not) students in solving them. I end with a discussion of conclusions and implications of this study. Background Learning a Second Language English language learners face the dual task of learning new linguistic structures in the second language and learning academic content. The process of learning this new linguistic system and the academic content does not occur automatically. In fact, to attain the same performance as native English speakers in the content areas, ELLs may take five to seven years or more to develop academic proficiency in a second language (Collier, 1995; Cummins, 1986). Cummins (1992) distinguished between two levels of language proficiencybasic interpersonal communication skills (BICS), i.e., social language, and cognitive/academic language proficiency (CALP). If students have not developed the language used for academic tasks in their first language (L1), they may experience difficulty with CALP in their second language (L2). Constructing Meaning: ELLs and Mathematics 5 Cummins' description of the need for cognitive/academic language proficiency (CALP) in Ll before L2 has two implications. One is that if the bilingual student has CALP for mathematics in Ll, then mathematical concepts (i.e., addition, subtraction, division, multiplication) attained in L 1 will transfer to L2. What changes for students are the lexical items (vocabulary) attached to these concepts. However, reading word problems will be significantly complex for two reasons. One involves the mental time it takes to process linguistic structures from one language to another. There may be words that do not make sense to the reader. These two can lead the reader to misinterpret text. Second, mathematics word problems are not straightforward either (Pimm, 1987; Ron, 1999). To encourage CALP in L2, students need mathematical activities that address language in natural contexts as well as mathematical contexts. "Initial reading should focus on meaning" (Miramontes, Nadeau, & Commins, 1997, p. 133) so that students learn to differentiate between natural language and mathematical language. Secondly, according to Cummins, if the student does not have CALP in Ll, the student will have considerable difficulty developing CALP in L2. The use of LI by teacher and the student may be very necessary in order to clarify confusions that result because of differences between English and Spanish mathematical terms (Cuevas, 1984) in mathematical problem solving. Mathematical Problem Solving Using Mayer's model (1987) of Problem Solving, Cardelle-Elawar (1995) defined four types of processes required to solve mathematics word problems. The first process involves translating, which requires two types of knowledge. The first type of knowledge is linguistic knowledge, which allows students to understand English sentences and translate from words to numbers or symbols. Using schema theory to understand different 6 Constructing Meaning: ELLs and Mathematics 6 ways ESL students may interpret (or misinterpret) texts (Alexander, Schallert, & Hare, 1991; Anderson & Pearson, 1984; Schallert, 1991) is critical in the first process of problem solving. The second type of knowledge is factual knowledge, which requires that students know certain facts in order to make a mental representation of a problem. For example, factual knowledge needed in order to convert 60 centimeters to 0.6 meters would involve knowing that 100 centimeters equals one meter. A second process defined by Cardelle-Elawar (1995) involves integration, in which students are required to use schematic knowledge in order to recognize different problem types. The third process involves planning and monitoring. The students are required to create a plan to solve the problem by breaking the problem into subproblems and establishing a sequence for the solution. Finally, the fourth process is solution execution. Students are required to use procedural knowledge to apply the rules of arithmetic accurately and efficiently while carrying out the calculations. Overall, Mayer's (1987) model has implications for the processes that are involved in problem solving, especially for non-native English speakers. The first step, translation, is a very critical step since students cannot proceed with the problem solving process unless they can use their own words to understand the problem. Some of the difficulties experienced by ELLs during the process of solving word problems are discussed next. Mathematics Register A register, as defined by Romaine (1994), "is typically concerned with variation in language conditioned by uses rather than users and involves consideration of the situation or context of use, the purpose, subject-matter, and content of the message, and the relationship between participants" (p. 20). Hence, a register is what you are speaking, 7 Constructing Meaning: ELLs and Mathematics 7 i.e., determined by what you are doing (nature of activity in which language is functioning) (Halliday & Hasan, 1989). Registers tend to differ in semantics and, therefore, in grammar and vocabulary. In addition, as Khisty (1995) explained, the mathematics register involves the nature of the language used to communicate mathematical ideas. However, the mathematics register should not be limited to the special terminology used in mathematics. Rather, the mathematics register extends to the use of natural language in a way that is particular to a role or function (Halliday, 1978, as cited in Khisty, 1995, p. 282). There are many ways a mathematics register can be developed, including reinterpreting existing words in natural language such as "point", "reduce", "carry", "set", "power", and "root". Thus, there are many words in the mathematics register which have different meanings from what children initially expect. In fact, Cazden (1979) stated that mathematics tends to be a restricted language, "limited in both size and meaning" (p. 135), by which she meant that mathematical terms are rarely encountered in non- mathematical contexts. This has important implications for ESL students learning mathematics. Researchers (Carrasquillo et al., 1996; Kang, 1995; Kessler, 1986, 1987; Khisty, 1992, 1995; Olivares, 1996) identified areas that create difficulty for ESL students when engaged in problem solving. Overall, these areas include vocabulary, syntax, and discourse. Vocabulary. The role vocabulary plays in the learning of mathematics by second language learners cannot be underestimated. As explained above, there are words used in natural language that take on new meaning in mathematics. For example, words such as "equal," "rational," "irrational," "column," and "table" would need to be explained to ELLs in the 8 Constructing Meaning: ELLs and Mathematics 8 mathematical and non-mathematical contexts. These two different explanations are needed for two important reasons. One is that ELLs tend to acquire the basic communication skills during the first two years, thus some may already know the everyday usage of the vocabulary but not the mathematical meaning. Secondly, explaining the mathematical meaning increases students' opportunities to acquire CALP, the language needed to perform in the content areas. In addition, words that indicate the same mathematical meaningi.e., "add," "plus," "combine," "and," "sum," and "increased by"also need to be emphasized to students by using them repeatedly in different word problems. Research has shown, however, that teachers do not focus much on vocabulary when teaching second language learners (SLLs) (Khisty, 1992). The instruction that is given is not thorough in its explanation of basic concepts. Therefore, it is essential that mathematics educators place emphasis on new vocabulary when introducing a new concept. If possible, the walls should be covered with charts that stress important words (Buchanan & Helman, 1993). Syntax. When reading word problems, second language learners may experience difficulties with syntax, which involves using correct language. Some examples include comparatives such as "is less than," "is greater than," and "x times as much." For example, consider the following problem: Write an equation using the variables S and P to represent the following statement: 'There are 6 times as many students as professors at this university. Use S for the number of students and P for the number of professors.' (Mestre, 1988, p. 208) Constructing Meaning: ELLs and Mathematics 9 In this word problem, 35% of nonminority engineering undergraduates provided an answer of "6S=P," reversing the order of the variables (Clement, Lochhead, & Monk, 1981, as cited in Mestre, 1988). In comparison, a population of Hispanic engineering students made the same error with a frequency of 54% (Mestre, Gerace, & Lochhead, 1982, as cited in Mestre, 1988). During clinical interviews, the authors noted that students were aware that there were more students than professors. However, two explanations were given why students made this mistake. One was that S and P were treated as labels for "students" and "professors," instead of treating them as variables to represent the number of students and the number of professors. Secondly, students used a sequential left-to-right translation of the word problem. Thus, "six times as many students" was translated as 6S, and this was equated to P, the number of professors. In addition, Castellanos (1980) illustrated examples that lack one-to-one correspondence between mathematical symbols and the words they represent. One example is division: "23/7" is often read as "7 goes into 23." In many Spanish-speaking countries, however, the two numbers are transposed, and the problem is referred to by saying, "Vamos a dividir 23 entre 7," which literally translated means, "Let's divide 23 into 7." Thus, this can lead to a misinterpretation. A second example involves substitution. If in English we say, "Let's substitute 8 for x," we mean that in place of x we are going to use 8. If this is translated literally into Spanish, it will read, "Vamos a sustituir 8 por x," and will mean that in place of 8 we are going to use x, which is exactly the opposite. In algebra, a typical mistake involves mixing the order of the following: The number a is five less than the number b. This translates into symbols correctly as "a = b - 5," not "a = 5 - 1 0