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.

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Mathematics as a Second Language

Mathematics Words and Expressions

Transmission of knowledge requires language or symbolic representations. Most fields of knowledge require language for its communication. Each domain uses words, expressions, phrases and collections of words that have specific and contextual meanings. Mathematics, too, has special symbols, words and phrases, as well as everyday words, which are used in particular ways. It has its own vocabulary, syntax, and its structure.

At times, major gaps exist between a child’s language outside the class and the abstract symbolism and language of school mathematics. Because of the complexity of the mathematics language and because mathematics is a second language for most children, many experience difficulty in learning it.

The size of a child’s mathematics vocabulary is highly correlated with mathematics achievement. Mathematics language and concrete models help construct the corresponding conceptual schemas in the child’s mind. Through the linguistic component (lexical entries—language containers) the child understands, uses, and retains the mathematics concepts. Most words used for learning mathematics are derived from the native language of the learner (e.g., add) and many are carried over into mathematics from other languages (e.g., trigonometry). Mathematics vocabulary consists of five kinds of words or expressions:

  • Words that have the same meaning in mathematics and outside of mathematics. For example, circle, add, cylinder, etc. For these words, students already have conceptual schemas and experiences. In other words, they bring some ideas related to these words. When these words are used in mathematics, students can expand the existing schemas and meanings to accommodate the use of these words in mathematics. For this reason, these words and related mathematics are easily accessible to students.
  • Words that have different meanings in mathematics and outside. For example, distribute in common language means division, but distributive property over multiplication a(b + c) = ab + ac means multiplication by a to the sum of b and c. There are many words of this type: and, or, table, fraction, tangent, sum, product, slope, mean, mode, median, etc. Because these words have specific meanings in mathematics, the teacher needs to create new conceptual schemas either by setting patterns, concrete modeling, analogies, or reason. Relying on the meaning and usage of the word from the native language may create difficulties and misconceptions.
  • Words that are unique to mathematics and do not exist outside of mathematics. For example, exponent, quotient, numerator, denominator, trigonometric ratios (such as sine, cosine), conic sections (such as hyperbola), differentiation, integration, quadratic equation, scale, figure, etc. These words are new to most students, so a teacher must take great care in their introduction to students. Concrete modeling and practice are the best ways to introduce them. For example, terms such as conic sections (circle, ellipse, parabola, hyperbola) are best understood when students actually make sections from the cone.
  • Compound words that are formed using competing and even opposite meaning. Words and expressions in this category include: least common multiple, greatest common factor, simultaneous linear equations, etc. For example, the term Least Common Multiple of numbers of 4 and 6 is best understood when we compare this expression with the statement: John is an intelligent, handsome, tall boy. The words intelligent, handsome, tall are qualifiers of the word boy and are to be seen in a certain order. Similarly, Least Common Multiple of 4 and 6 must be seen in a certain order. The order of the activity is: first we find the multiples of 4 and 6. Then we mark (the numbers in bold) the common multiples of 4 and 6. Lastly, we select the least of these common multiples (12). Therefore, the Least Common Multiple of 4 and 6 is 12. It is important to emphasize the interaction of the words in the expression Least Common Multiple. The same process applies in the case of Greatest Common Factor of 20 and 24.

4     8   12   16   20   24   28   32   36    40   44   48, …

     6     12      18       24      30        36        42       48, …

  • Compound words formed by combining different concepts. These include: miles per hour, scale factor, density, etc. Before understanding these types of words (compound secondary concepts), students need to have a clear understanding of the corresponding component primary concepts involved in the secondary concept.

Just like in the native language, every word may have different meanings according to the context. For example, a word in the English language may be understood and its meaning found by examining its (a) entomological roots, (b) historical usage, (c) popular meaning, (d) contextual, and (e) meaning received by the listener or reader. Mathematics words, most of the time, are more consistent in meaning than they are in the native language. However, the same approach to examining the roots and history of the word may help us with the meaning of the word or expression in the context of mathematics.

Mathematics Words and Expressions

Numeracy: Learning Difficulties

Numeracy is the ability to execute standard whole number operations or algorithms correctly, consistently, and fluently and estimate, accurately and efficiently, both mentally and on paper, using a range of calculation strategies and means.

Acquisition of numeracy depends on proficiency in numbersense – the flexible use of number relationships and numerical information in various contexts. Those with numbersense can represent and use a number in multiple ways depending on the context and purpose. In computations and operations, they can decompose and recompose numbers with ease and fluency.

Numeracy is the gateway to higher mathematics beginning with the study of algebra and geometry. But, many individuals never reach this gateway because of difficulties with mastering numeracy. Some difficulties are caused by environmental factors:

  • lack of appropriate number experiences,
  • ineffective instruction and a fragmented curriculum,
  • inefficient conceptual models and strategies,
  • lack of appropriate skill development, and
  • low expectations.

For example, teaching arithmetic facts by sequential counting (“counting up” for addition, “counting down” for subtraction, “skip counting” on number line for multiplication and division), though advocated by many researchers and educators, is not an efficient strategy because one-to-one counting only turns most children into counters – that’s all.

Other difficulties in mastering numeracy are due to individual capacities and learning disabilities. Among those who exhibit learning problems in mathematics, some struggle in certain aspects of mathematics (e.g., procedures, conceptual processes). Some have difficulty in arithmetic, algebra, or geometry. Others display symptoms of dyscalculia, which manifests as poor number concept – difficulty in estimating the size and magnitude of numbers, lack of understanding and fluency in number relationships, and inefficiency of numerical operations.

A learning disability may manifest as deficits in the development of prerequisite skills: following sequential directions, spatial orientation/space organization, pattern recognition and extension, visualization and visual perception, and deductive and inductive thinking. These deficits may affect learning in different aspects of mathematics, for example, a few isolated skills in one concept or procedure or several areas of arithmetic and mathematics. Some learning problems fall in the intersection of quantity, language, and spatial thinking.

Because of the range of mathematics disabilities, we cannot always identify a cause or effect, and no one explanation adequately addresses the nature of learning problems in mathematics.

Numeracy: Learning Difficulties

Numeracy

Numeracy, executing mathematics procedures fluently, precisely, and consistently with understanding is like reading fluently with comprehension. Reading comprehension is essential to access information from text. Numeracy and literacy are similar. Both processes engage us in seeking and making meaning. Mouthing words without comprehension is of little value. Similarly, recalling facts and executing a mathematics procedure, even fluently, without understanding the purpose and meaning of the procedure is of little value because it does not take us far in mathematics.

When reading, we need to know the contextual meaning of words and the objective of reading a passage. Similarly, in the context of mathematics, it is important that we know the answers to questions such as: Is the right procedure used here? What does this step in this procedure accomplish and why? Where does this step come from and what is its role in the procedure? Why does this procedure work in this problem? What do I expect as the answer? Am I right? Is this the most efficient way of doing this problem? Can I apply this method in different situations? Under what conditions does this procedure not work? Etc. The ability to answer these questions shows an understanding of mathematics concepts.

Answers to these questions and knowing the reasons behind procedures and the language and concepts associated with the procedures are essential for being a numerate person. Mastery of numeracy skills—fluency in the execution of the four arithmetic operations on whole numbers with precision, consistency and with understanding is like being able to read fluently with comprehension, being literate.

The statement learning to read by the third grade and then reading to learn defines the standards of competence in reading. Reading here means reading accurately, fluently, and with comprehension. To me, it has its counter part in mathematics: mastery of numeracy by the fourth grade and then using numeracy to learn mathematics. Numeracy means acquiring efficient and accurate strategies that lead to successful solutions of problems. Numeracy makes meaning of numbers and their relationships. Being numerate means (a) having the necessary mathematics concepts and skills to meet common quantitative and spatial demands, (b) having the confidence and intuition to apply quantitative principles in everyday life, and (c) being able to use these skills appropriately in diverse settings. In sum, being numerate requires the knowledge and disposition to think and act mathematically.

 

Numeracy