Mathematics as a Second Language   For many students, achieving proficiency in reading and writing in their native language is a difficult task. The use of language in content specific disciplines such as mathematics makes that task even harder. In fact, students need considerable proficiency in the native and mathematical language because in a mathematics classroom, they need to constantly translate between mathematical and everyday language. For this reason, the framers of the Common Core State Standards (CCSS-M, 2010) recommend that schools and publishers of mathematics textbooks provide a high degree of familiarity with words, syntax, and grammar, as well as styles of presentation and arguments that are not part of informal talk. Many people perceive mathematics learning as difficult, and increased linguistic demands heighten this perception. Effective teachers have always recognized language as an essential tool for the conceptualization of mathematics. The language used to convey mathematical information in teaching and learning is important because:

• The number of students in our schools with limited English language experiences or who speak English as a second language has increased dramatically.
• Students confront specialized and rigorous language in new mathematics textbooks.
• The concepts in reform curricula make new demands on students’ linguistic abilities and emphasize an enlarged vocabulary.
• The development of the conceptual schemas is not only facilitated by a strong linguistic component but also depends on it.
• The increased realization that mathematics achievement is directly related to a student’s mathematics language development, particularly the size of the mathematics vocabulary. Early and better performance in mathematics predicts achievement in higher mathematics as well as in reading.
• The modern technological society is built on strong numeracy as much as it is on literacy.

The Role of Language in Mathematics Learning The vital role of language in mathematics performance is widely recognized. Most people see mathematics language only in the context of word problems, but it is much more than that. Every mathematical idea involves three components: linguistic, conceptual, and procedural. To learn an idea means to create linguistic and conceptual models for it. We need to have a language container to receive, comprehend and explain a concept. Without an internalized language container, we need to relearn the concept every time we encounter it. The term language container means a word or a phrase to express an idea with related conceptual schema: sum, even number, least common multiple, denominator, rectangular solid, and conic section. Students’ proficiency in mathematics is directly related to the size of the set of their language containers. However, rote memorization of a collection of words is not enough to master the language of mathematics. One has to acquire the related schema with understanding. Mathematics uses special words and phrases and many everyday words in particular ways and with special meanings to describe phenomena and concepts. The difficulty for children lies in the gap that exists between their native language and the language of mathematics. For many, this gap is a barrier to learning and using mathematics. Poorly written textbooks cause some of the linguistic difficulties children have in mathematics. New technical terms crowd textbook pages. Explanations are unintelligible. New words and terms are introduced only as recipes, without adequate explanations and examples. Children’s mastery of most mathematical concepts is dependent on the interplay between language, concepts, and models. Mathematical conceptualization is independent of language in early childhood. However, once a child has acquired language fluency, mathematics and language interact. Progress in one enhances development in the other (except perhaps when there is a learning disability). Ability in mathematics is a manifestation of two different aspects: mathematical insight and knowledge and facility with language. Students who have difficulty with literacy often find themselves having similar problems with numeracy. For instance, almost 40 percent of dyslexics also show symptoms of dyscalculia. Children from cultural and social environments with little or no emphasis on numeracy are not well prepared to learn formal mathematics as they lack language containers for these concepts. They have a backlog of numeracy learning to catch up. In contrast, children exposed to quantitative and spatial representations bring prior knowledge about quantity and space to school. Children with facility in their native language, even if it is different from the language of instruction, are better prepared for numeracy. If they possess the language containers for concepts in their native language, they can develop language and conceptual schemas in the second language. Acquiring the Mathematics Language Mathematics is a second language; it has its own alphabet, symbols, vocabulary, syntax, and grammar. Numeric and operational symbols are its alphabet; number and symbol combinations are its words. Equations and mathematical expressions are the sentences of this language. Mastery of a mathematical concept is the result of an interactive process between language and quantitative and spatial experiences. Initially, concrete experiences with quantity and space form concepts and are communicated through visual representations and artifacts. Later, children learn to represent them symbolically/abstractly. Abstract symbols, formulas and equations are then applied to solving problems. This iterative and cyclic process is called mathematization. The various linguistic activities serve different purposes in developing conceptual schemas and the acquisition of mathematical procedures and skills. To develop a mathematics language we need: Vocabulary and symbols, Syntax, and two-way translation. Vocabulary and symbols: words, terms and symbols can represent a complex concept. Comprehending the statement of the problem (the terms and words involved) and understanding the intent of the problem (what concept and procedure is involved in the problem) requires a student to have a strong vocabulary and associated conceptual schemas. For example, on a recent state examination, some students did not answer the problems (Find the sum of 8.7 and 5.2. Find the product of 1.2 and 1.3.) because they did not know the meanings of sum and product. Every word has at least five meanings: (a) epistemological—the origin of the word, (b) historical—the meaning acquired over time, (c) intended meaning, (d) current meaning, and (e) meaning received and understood by the reader (as Marshall McLuhan said: “message received is message sent.”). To truly understand the meaning of a word, one needs to understand as many meanings of the word as possible.  Words are language containers for ideas and concepts. We cannot have a concept, if we do not have a language container for the concept. Similarly, a word has no value for a person, if he does not have a concept behind the word.  Understanding a word means that person has an associated schema with the word and can also use it. Comprehension in reading and understanding in mathematics come when the child possesses the conceptual schemas behind words. This is particularly so if one wants to help children learn a second language and the meanings of words and expressions in the language. The challenge becomes more complicated when children are learning a second language like mathematics where every word is packed with complex concepts and schema.  Syntax: organization of words and structure of mathematical expressions. Some children’s mathematics difficulties are due to not understanding the order of words in a sentence. For example: the difference between ‘subtract 5 from 3’ or ‘subtract 3 from 5.’ .75 divided by .89 or .89 divided by .75. Translation: translating from mathematics sentences into English and from English into mathematical expressions. When students encounter a word problem, many ask the teacher to supply the operation involved in the problem. Once the teacher provides the operation, the students perform the appropriate operation and give the answer. Often, students solve a problem but do not know what the answer means. Both of these examples are problems of translation. To be proficient in mathematics, students have to navigate between mathematics and native languages.