Mathematics and ESL - Center for Teaching and Learning

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tural differences in the symbolic denotations of mathematical processes. Also, while certain words designate specific math operations, (“less than” may indicate subtraction), the opposite may also be true: Hilda has 7 records. She has 5 records less than David. How many records does David have? Since David has 7+5, “less than” indicates addition. (Chamot and O’Malley 229). Syntactically, students must understand that they cannot read mathematical sentences as if they were natural language sentences. In algebra, for example, Crandall, Dale, Rhodes and Spanos found “recurring errors in translations of the language of word problems into solution equations,” (Dale and Cuevas 334), indicating that students map the surface syntax of the problem statements onto their equations. Students translated the sentence, “The number a is five less than the number b” as a = 5 - b, instead of the correct translation a = b - 5. (Dale and Cuevas 335). Similarly, given the problem, “There are 6 times as many students as professors in the math department. Write an equation that represents this statement,” students typically write 6S=P because they follow the literal word order of the natural language sentence (Cocking and Mestre 216). The frequent use of complex structures which are traditionally taught at higher levels of ESL, such as comparatives: “That car costs 5 times as much as mine does,” passive voice: “X is defined to be equal to 0” (Dale and Cuevas 334), and logical connectors which set up various relationships between parts of a text – including cause-effect, similarity, and logical sequence: “if…then, if and only if, such that...” – also contributes to the level of linguistic difficulty. Semantically, to understand mathematical language, students must be familiar with discourse devices which signal reference in English. In the example provided by Dale and Cuevas – “The sum of two numbers is 77. If the first number is 10 times the other, find the number.” – students need to understand that there are 2 numbers, that “the first number” refers to one of those numbers, and the elliptical “the other” refers to the second number, whose value they have to find (336). What makes the language of math so difficult, then, is both the students’ lack of a cultural framework on which to build an understanding of the context of a problem, and their lack of familiarity with the discourse features of math texts. This brief description of the challenges inherent in solving mathematical word problems reflects the interactive nature of the learning process itself. Parallels Between Second Language and Mathematics Learning Language learning and mathematics learning are both cognitive processes. They can be understood through a constructivist model in which meaning is derived through the interaction between one’s own background knowledge and experience (top-down processing) and one’s ability to process the given task or decode text (bottom-up processing). These two processing modes are complementary; one can be used to compensate for weakness in the other, but each alone is insufficient in terms of successful learning and performance. They both develop through feedback. Top-down processing refers to higher-order thinking skills. In mathematics, this is revealed in conceptual understanding of structures and patterns, appropriate application of basic arithmetic and algebraic operations and concepts. Similarly, in language learning, this translates as the activation and application of appropriate background knowledge to the creation of meaning. However, students capable of this level of understanding still need to rely on their bottom-up processing skills to ensure that their hypotheses are grounded in the details of the text. Bottom-up processing refers to lower level skills that must be practiced if automaticity is to be achieved. In mathematics, as in language acquisition, this is associated with rote learning: computational skills, memorizing facts and formulas, doing mechanical, de-contextualized arithmetic or grammar drills, and focusing on 2 • In Transit
discrete elements often without the benefit of comprehension. Students who remain at this level of processing cannot be said to be thinking mathematically or using language for authentic communicative purposes. This view of learning is process rather than product-oriented. It implies an explicit focus on learning strategy use and self-monitoring if learning competence is to be transferable. It implies actively listening to students before attempting to teach, taking into account their schema knowledge and expectations of how things work. It is incumbent on teachers to elicit and interact with students’ prior understandings, to mediate their difficulties so that a restructuring of students’ knowledge may take place. The Common Methodological Debate In both ESL and Mathematics classrooms, deductive teaching clearly prevails. Students are repeatedly taught the rote skills of computation or grammar, despite disappointing results. The debate over the merits of a deductive versus an inductive approach goes on. Should teachers begin by building bottom-up skills in the hope that top-down, conceptual, procedural understanding will somehow magically follow? Or should they begin by ensuring the development of top-down processing while supporting the development of the necessary bottom-up skills? Uncommon Solutions Trends toward content and task-based methodology in ESL indicate a call for a departure from traditional deductive teaching. In mathematics, a statement in the Principles and Standards document published by the National Council of Teachers of Mathematics in 2000 claims, “Solving problems is not only a goal of learning mathematics but also a major means of doing so” (Van de Walle 40). We argue that inductive, problem-based approaches provide a more authentic context for learning that is not separated from doing. These approaches are more learner-centered and engaging and provide the window into students’ math and language hypotheses that teachers need in order to skillfully determine the sources of student error. Teachers can then provide feedback at that crucial moment when a meaningful revision of hypotheses is apt to occur. We demonstrated this during our workshops with two problem-solving activities used to initiate lessons, one from an ESL grammar book (Badalamenti and Henner Stanchina 98) and one reported by Ball and Bass (91–94). By asking ESL students to study a picture of an apartment, decide who lived there, and justify their conclusion, the teacher was able to elicit numerous sentences showing a range of hypotheses on the structure: There is / there are, including “in the bedroom has dress,” “is a dress,” “there are have computer,” “there is the computer,” and “there is a coffee pot on the stove.” Similarly, by presenting the following problem: “Joshua ate 16 peas on Monday and 32 peas on Tuesday. How many more peas did he eat on Tuesday than he did on Monday?” (Ball and Bass 91), the teacher elicited six different problem-solving methods. The challenge for both the ESL and the Mathematics teacher lies in creating the task, identifying and interpreting student errors, using them to understand student learning, and making decisions about teaching. The advantage of this methodology for the learner is a more connected, deeper understanding. This approach also lends itself to a focus on writing to learn. The Principles and Standards for School Mathematics support communication as an “essential part of mathematics and mathematics education” (NCTM 60). The standards encourage language use, particularly for second-language learners, in order to facilitate “communicating to learn mathematics and learning to communicate mathematically” (NCTM 60). There are many ways teachers can help students integrate language development with the learning of mathematics. These include having students paraphrase word problems, create word problems, justify solutions, and explain procedures for problem solving. We plan to pursue our conversation about the relationship between language and math- Mathematics and ESL • 3
Page 1: Mathematics and ESL: Common Ground

Mathematics and ESL - Center for Teaching and Learning

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