We cannot receive, hold and manipulate a concept without having either an image/schema or words for the concept. The image could be a picture, figure, drawing, or symbols. In the case of language it can be a word, expression, or an equation.

For effective communication of mathematics ideas, children need robust and rich images and vocabularies (language containers). Without appropriate language containers, children cannot retain and communicate mathematics ideas. Vocabulary—words, expressions, phrases—are the language containers for mathematics concepts.

Learning mathematics, then, is using, creating, extending, and modifying language containers—the vocabulary of mathematics. Students’ proficiency in mathematics is directly related to the size of the set of their vocabulary. Rote memorization of a collection of words is not enough to master the language of mathematics. Instead, one has to acquire the related schemas with understanding. *Language proficiency* refers to the degree to which learners exhibit control over their language.

The introduction of mathematics vocabulary and terminology should be contextual, but even direct study of quantitative and spatial vocabulary contributes significantly to improved mathematics conceptualization—learning new concepts, creating deeper and robust conceptual schemas, and more effective communication.

When children create and encounter a language container for a mathematics concept, they also create and invoke the related conceptual model in their minds. Each word and expression such as *sum*, *product*, *rational number,* *least common multiple,* *denominator*, *rectangular solid*, *conic section*, and *asymptotic *represents a concept with its related schema*.* For example, if a person understands the definition of multiplication as ‘repeated addition’ or ‘groups of’, then these expressions invoke the conceptual schema. The expression 43 ´ 3, will invoke: 43 repeated 3 times (43 + 43 + 43) or 3 groups of 43 (43 + 43 + 43). If multiplication is learned as the ‘area of a rectangle’, then 3 ´ 43 will invoke an image of a rectangle with dimensions 3 (vertical side) and 43 (horizontal side).

The development and mastery of mathematical vocabulary are the result of a long and continuous interactive process between native language, mathematics language and symbols, and their quantitative and spatial experiences. This begins with play and concrete experiences in children’s environment. Experiences are represented through pictorial and visual forms and means, which then may result in abstract mathematics formulations and problems that students solve. This mathematics formulation—devising of abstract symbols, formulas, and equations, is then applied to more problems, and the result of this process is communicated. Successful communications demonstrate that the child has mastered a concept. The process can be summarized as:

- Understanding the environment (concrete experiences and use of native language).
- Translation (native language to pictorial and linguistic forms).
- Representation (in the native language).
- Description and verbalization (in the native language).
- Discussion (in the native language).
- Mathematical formulation of the problem (in the mathematical language).
- Manipulation of mathematical language.
- Communication of the outcome of mathematics operations (in mathematics and native languages).

This communication furthers not only children’s mathematics achievement but also their language development.

**Building the Vocabulary of Mathematics
**Many of children’s mathematics difficulties are due to their limited vocabulary—its size, level, and quality. A child’s size and level of vocabulary is the intersection of three language sets:

- The level and mastery of the native language and background the child brings to the mathematics task.
- The level and sophistication of language that the teacher uses and the questions she asks to teach mathematics.
- The language set of the mathematics textbook being used.

The intersection of these three language sets is the available language the child has to learn mathematics. A small intersection means the child has a limited vocabulary. The objective, then, is to increase the size of this intersection. A child’s limited mathematics vocabulary may be for many reasons.

- The mathematics problems of the child with English as a second language in a classroom where the medium of instruction is other than the child’s native language.
- The child’s and teacher’s economic, cultural, and geographical backgrounds differ. For example, the linguistic problems that many urban black children and immigrant children face are an example of a linguistic/cultural mismatch and the assumptions teachers make in instructing children.
- Textbook language sets differ from the language sets of the children and the teacher.

Whatever the reasons for limited language sets, we need to help children acquire a robust mathematics vocabulary. Properly acquired and used in context, a mathematics vocabulary has a profound effect on children’s mathematics achievement and their thinking. Planned activities for developing, expanding, and using vocabulary contribute significantly to better mathematical word problem-solving ability and support learning new concepts, deeper conceptual understanding, and more effective communication.

Although more textbooks are emphasizing the language of mathematics, there is still little attempt to develop a coherent and comprehensive mathematics vocabulary in school mathematics teaching. In one textbook, the expression “*find the sum*” is introduced quite early. In another series, the expression is introduced much later, and then the words “find the sum” and “add” are used interchangeably. In another text, the word “sum” is used sparingly. Consequently, a child may face different language sets from grade to grade and from school to school. Although the textbooks have a large number of common language terms and vocabulary, many words are not in common. Further, some textbooks use so much language without properly introducing the terms that many children find textbooks frustrating. Exercises do not provide enough practice in basic skills, which prevents children from automatizing the language or the conceptual skills associated with them.

**Strategies for Enhancing the Mathematics Vocabulary
**Ways in which children’s failure to develop mathematical vocabulary may manifest as: (1) children have difficulty conceptualizing a mathematics idea; (2) they do not respond to questions in lessons; (3) they cannot perform a task; and/or (4) they do poorly on tests, particularly on word problems.

- Their lack of conceptualization of a mathematical idea may be because they do not have the language for the concept to receive it, comprehend it or express it, such as ‘find the sum of’, ‘union of two rays…,’ ‘evaluate…’
- Their lack of response may be because they do not understand spoken or written instructions such as ‘draw a line between…’, ‘touch the base of the triangle’, ‘place a positive sign next to the numeral,…’ or ‘find two different ways to…’
- They are not familiar with the mathematics vocabulary words such as ‘difference’, ‘subtract’, ‘quotient’, or ‘product.’
- They may be confused about mathematical terms such as ‘odd’ or ‘table’, which have different meanings in everyday English and have more precise meanings in mathematics.
- They may be confused about other words and symbols like ‘area’ and ‘perimeter’, ‘factor and multiply’, ‘and’.

To enhance children’s vocabulary, every school system should have a minimal ** mathematics vocabulary list** at each grade level. Mastery of words from such lists will prepare children to communicate mathematics. This list can also be used to assess students’ grade level language of mathematics. This list should indicate the grade of introduction of words, terms, and definitions and the level where they are mastered. It should be developmentally and linguistically appropriate. The teacher should constantly identify, introduce, develop, and display the words and phrases that children need to understand and use.

The teacher should use the same techniques to introduce mathematics words as she teaches native language. She should have a ** Math Word Wall** for every mathematics concept she teaches. When a new word related to the concept emerges in discussion, it is added to the Word Wall. With the introduction of each word, students are exposed to several words and concepts that contain it. Then students use it in their own words, with as many examples as they can. The teacher selects a word and then asks children to use it in mathematics context. The following exchange illustrates this process.

**“**Give me a sentence that uses the word ‘add.’”**“**You have $5 and I have $14. Let us add both amounts.”**“**That is great! Now use the word ‘sum’ in a sentence.”**“**That is easy. If we add our monies, what is the sum of our monies?”**“**That is great! Now I am going to write some words on the board. I want you to first to tell me and then write a sentence or two using each word. If you want, you can use more than one word in a sentence.”

The concepts are then reviewed in circular fashion, built upon, and tied into new ideas. This helps children construct a working vocabulary that is constantly augmented, and they are also learning skills to build it.

Once the key root words have been introduced to children, the teacher can begin to extend the mathematics vocabulary words. Among the easiest sets are the words formed with prefixes, suffixes and derivative words. The process is to introduce the math prefixes and roots casually and then formally. In a casual manner, parents and teachers can remark, “You know a tricycle has 3 wheels. Tri- means 3 and cycle means wheels.”

**Teacher: **What will be the name of the object that has three angles?

**Student: **A triangle.

**Teacher: **Why?

**Student: **A triangle has 3 angles and tri- means 3.

**Teacher: **Now draw a triangle on your paper.

Children draw triangles on their papers.

**Teacher: **The word ‘lateral’ means a side. What will you call an object that has three sides?

**Student: ** A trilateral.

**Teacher: **Now draw a trilateral on a paper.

Children draw a trilateral on their papers.

**Teacher: **If the word ‘gon’ means a corner, what will you call an object that has three corners?

**Student: **A trigon.

**Teacher:** If ‘octo’ means eight, what does ‘octagon’ mean?

**Student:** A figure with eight corners.

As with all language development, there is a sequence in moving from speech ability to writing ability: the input is auditory in its foundation (the child is immersed in oral linguistic experiences), then followed by speech ability (the child produces language) and later by reading and writing ability. When young children have this kind of foundation, they avoid the anxiety of making sense of key foreign words later on in a formal setting. They will be able to generalize and relate math concepts to their daily experiences.

**Instructional Suggestions for Language Proficiency
**There are practical reasons children need to acquire rich and appropriate vocabulary for them to participate in classroom life—the learning activities and tests. There is, however, an even more important reason: vocabulary, as part of mathematical language, is crucial to children’s development of thinking not only in mathematics problem solving but in general problem solving. Once children have control over their language usage, they begin to have control over the meta-cognitive skills that produce insights into their learning and their interactions with learning tasks. Language and thinking are interwoven in reasoning, problem solving, and applications of mathematics in multiple forms—intra-mathematical, interdisciplinary, and extracurricular. If children do not have the vocabulary to talk about a concept, they cannot make progress in understanding its applications—therefore solving word problems.

Teachers often use informal, everyday language in mathematics lessons before or alongside technical mathematical vocabulary. This may help children’s initial grasp of the meaning of words; however, a structural approach to the teaching and learning of vocabulary is essential to move to higher mathematics using the correct mathematical terminology. This also applies to proficiency. The teacher needs to determine the extent of children’s informal mathematical vocabulary and the depth of their understanding and then build the formal vocabulary on it.

It is not just younger children who need regular, planned opportunity to develop their mathematical vocabulary. All students and adults returning to education need to experience a cycle of concrete work, oral work, reading, writing, and applications.

The teacher needs to introduce new words through a suitable context, for example, with relevant, real objects, mathematical apparatus, pictures, and/or diagrams. Referring to new words only once will do little to promote the learning of mathematics vocabulary. The teacher should use every opportunity to draw attention to new words or symbols with the whole class, in small groups or with individual students. Finally, the teacher should create opportunities for children to read and write new mathematics vocabulary in diverse circumstances and to use the word in sentences.

Concrete materials/models develop images and the language for mathematics ideas. The concrete materials/models help children (a) generate the language, (b) understand the concept, and (c) arrive at an efficient procedure. Students should be encouraged to explore and solve problems using manipulative materials and asked to discuss and record the activity using pictures and symbols. The teacher or a student can also act the word out.*Concrete work:*The teacher should explain the meanings of words carefully. The teacher should refer to a similar word; give the history and the derivation of the word and write it on the board. Children should copy it in their Math Notebook. The teacher should ask the children to say the word clearly and slowly. They should rehearse the pronunciation of the word. The teacher should ask them to spell the word and ask a child to say the word and spell it with eyes closed.*Writing work:*Students describe the work done at the concrete level, using mathematics words and expressions based on the visual and tactile experience of the meaning of mathematical words in a variety of contexts. This oral work may be facilitated in different contexts by*Oral work:*- listening to the teacher or other students using words correctly
- acquiring confidence and fluency in speaking, using complete sentences that include the new words and phrases, in chorus with others or individually
- discussing ways of solving a problem, collecting data, organizing data and discussing the properties of the data for a variety of reasons: to generate hypotheses, develop conjectures or make predictions about possible results or relationships between different elements and variables involved in the problem
- presenting, explaining, communicating, and justifying methods, results, solutions, or reasoning, to the whole class, a group, or partner
- generalizing or describing examples that match a general statement
- encouraging the use of the word in context and helping sort out any ambiguities or misconceptions students may have through a range of open and closed questions.

Because students cannot learn the meanings of words in isolation, I believe in the centrality of reading and conversation in mathematics lessons. Shared reading is a valuable context for learning and teaching not only mathematics language but also mathematics content. Strategies such as using children’s books, stories, DVDs, and videos as a vehicle for communicating mathematical ideas develops mathematical language. Reading word problems aloud and silently, as a whole class and individually, is equally important. During these readings, the teacher should ask questions involving mathematics concepts. This develops strong mathematics language and understanding. Students can be asked to read and explain:

- numbers, signs and symbols, expressions and equations in blackboard presentations
- instructions and explanations in workbooks, textbooks, and other multi-media presentations
- texts with mathematical references in fiction and non-fiction books, books of rhymes, children’s books during the literacy hour as well as mathematics lessons
- labels and captions on classroom displays, in diagrams, graphs, charts, and tables
- definitions in illustrated dictionaries, including dictionaries that the children have made themselves, in order to discover synonyms, origins of words, words that start with the same group of letters (e.g. triangle, tricycle, triplet, trisect…), words made by coding pre-fixes or suffixes, words derived from other words.

All students from K through 12 and adults returning to education need to work on developing their mathematics vocabulary.