Mathematics As a Second Langauge

Mathematics As A Second Language: Reading, Writing and Doing Mathematics

Mahesh C. Sharma

Professor of Mathematics Education and President Center for Teaching/Learning of Mathematics Wellesley, Massachusetts

Blue Ribbon conference Reading Public Schools Reading, MA April 15, 2016

1 Introduction1

A mathematician is like a poet in practicing the economy of language.

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. For many students, achieving proficiency in language is a difficult task; use of language in content specific disciplines such as mathematics makes that task even harder. Current mathematics textbooks call for a high degree of familiarity with words, grammar, styles of presentation and arguments that are not part of informal talk. Children need to translate between mathematical and everyday language. The National Council of Teachers of Mathematics (NCTM, 2000) and Common Core State Standards in Mathematics (CCSS-M, 2010) recommend students read, write, discuss, and think in mathematics language. 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 a topic of interest because: . The number of students not well prepared in English, because of limited language experiences or English as a second language, has increased dramatically. . Students confront specialized and rigorous language in textbooks.

1 For in=depth treatment of the role of language in mathematics learning see: Dyslexia and other Language Related Mathematics Difficulties (Sharma, 2008).

2 . The conceptual components in reform curricula make new demands on students’ linguistic abilities and heavy emphasis on an enlarged vocabulary. Many people perceive mathematics learning as difficult; increased linguistic demands heighten this perception.

A. Role of Language in Mathematics Learning Most people see mathematics language only in the context of word problems, but it is much more than that. Mathematics uses special words and phrases and many everyday words in particular ways and with special meanings to describe phenomena and concepts. A gap exists between children’s language at home and the language of mathematics. For many children, this gap may create difficulty in learning and using mathematics. Some of the linguistic difficulties children have in mathematics are because of poorly written textbooks. Difficult new technical terms crowd textbook pages. Explanations are unintelligible. New words and terms are introduced without adequate explanations and examples and only as recipes. 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 vital role of language in mathematics performance is widely recognized. A positive correlation (.40 to .86) exists between mathematics achievement and reading ability. Children’s mastery of most mathematical concepts is dependent on the interplay between language, concepts, and models. Cognitive psychologists contend that 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 definite, specific 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. The incidence of dyslexia and dyscalculia is about 3% for each in general, but almost 40 percent of dyslexics also show symptoms of dyscalculia. Children exposed to quantitative and spatial representations bring prior knowledge about quantity and space and communication to school. Children from cultural and social environments where there is 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. Children with

3 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, then they can develop language and conceptual schemas in the second language. 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.

B. Mathematics as a Second Language Mathematics is a bonafide 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. Several models for the role language plays in the teaching and learning of mathematics have been presented earlier. A comprehensive model, outlining the use of language in mathematics conceptualizations, application, and problem solving, is presented here.

Native Language and Mathematics Language Interaction

(1) (2) (3)

PROBLEM REPRESENTATION STATEMENT OF THE IDENTIFICATION FROM PICTORIAL AND PROBLEM (WORD THE REAL WORLD LINGUISTIC PROBLEM)

(Native Language) (Native Language) (Native Language)

l a c i

t e g a g n

i m a e e k u v c h g i t e t n a a h a

C N M L (6) (4) (5) MATHEMATICAL DESCRIPTION AND (7) SOLVING DISCUSSION FORMUALTION SYMBOLIC MATHEMATICAL ANSWER(S) (EQUATIONS & REPRESENTATION TRANSLATION EXPRESSIONS) EXPRESSIONS (Native &Mathematical Mathematical Language Mathematical Language Language)

1) Understanding the environment (concrete experiences and use of native language). 2) Translation (native language to pictorial and linguistic forms). 3) Representation (in the native language). 4) Description and verbalization (in the native language).

4 5) Discussion (in the native language). 6) Mathematical formulation of the problem (in the mathematical language), and 7) Manipulation of mathematical language. Mastery of a mathematical concept by children is the result of a continuous interactive process between language and quantitative and spatial experiences. Beginning with concrete experiences with quantity and space, concepts are communicated through visual representations and artifacts. Finally, children learn to represent them symbolically/abstractly. Abstract symbols, formulas and equations are then applied to more problems, and the result of this process is communicated. This iterative and cyclic process is called mathematization. The model demonstrates that the various linguistic activities serve different purposes in the acquisition of mathematical concepts and skills. Students need considerable proficiency in the native and mathematical language to cope with the level of linguistic proficiency required for mathematics. 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. Some of children’s mathematics difficulties are due to their limited vocabularies. On a recent state examination, many students did not solve 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.

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. Similarly, 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 mathematical and native languages.

C. Building a Mathematics Vocabulary Direct study of quantitative vocabulary contributes significantly to improve mathematical conceptualization—learning new concepts, creating deeper and robust conceptual schemas and more effective communication. Many mathematics difficulties are due to children’s vocabulary—its size, level and quality. A child’s vocabulary level is the intersection of three language sets:

5 . The level of the language and background the child brings to the task. . The language that the teacher uses to teach mathematics. . The language set of the mathematics textbook. 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

6 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. The development of mathematics vocabulary, thus, is just like developing vocabulary in another language. 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. To enhance children’s vocabulary, school systems should have a minimal mathematics vocabulary at each grade level (e.g., The Mathematics Vocabulary List: Kindergarten Through Sixth Grade in Sharma, 2005). The mastery of words from such lists prepares children to participate in language communication. This communication furthers children’s language development. The teacher selects a word(s) from the list related to the content being taught, uses them contextually and helps children master them. For example, “Give me a sentence that uses the word ‘add.’” “You have $5 and I have $4. Let us add both monies.” “Now use the word ‘sum’ in a sentence.” “If we add our monies, what is the sum of our monies?” “Now I am going to write some words on the board. I want you to write a sentence or two using each word. If you want, you can use more than one word in a sentence.” The Mathematics Vocabulary List can also be used to estimate students’ grade level language of mathematics as it suggests the target grade for a particular word. Using some root word, the teacher can introduce prefixes, suffixes and derivatives. The teacher remarks: "A tricycle has 3 wheels. Tri- means 3 and cycle means wheels.”

7 “What will be the name of the object that has three angles?" “A triangle.” “Why?” “A triangle has 3 angles and tri- means 3.” “Now draw a triangle on your paper.” Children draw triangles on their papers. “If the word ‘nomial’ means ‘term,’ what do you think the word ‘trinomial’ means?” “Three terms.” “Great!” It is not just younger children who need regular, planned opportunity to develop their mathematical vocabulary. All students from K through 12 and adults returning to education need to work on developing their vocabulary. The Japanese Ministry of Education regularly publishes and publicizes commonly used vocabulary and when new words are coined to facilitate pedagogy and mathematics. Similarly in the United Kingdom, the Numeracy Project motivated by the National Curriculum has published a vocabulary list for use by teachers. 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. This

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

D. Use of Syntax and Voice in Mathematics Language The problem 1.4)56.43 is misread by more than 50 percent of middle and high school students. They read it as 1.4 divided by 56.43 rather than 56.43 divided by 1.4, or that 1.4 divides into 56.43. In a mathematical expression such as this, the order of words and symbols is important. Similarly, children find it easier to understand and express ideas and operations in the active voice. The express addition, subtraction, multiplication in the form ‘2 plus 4,’ ‘4 minus 1,’ ‘2 times 6,’ respectively. However, they have difficulty with division (e.g. 3.2 divided by 4). Because it is stated in the passive voice. Therefore, many children will write: 3.2)4 rather than 4)3.2 for 3.2  4. Limited language models also limit conceptualization—the development of conceptual schemas. For example, the problem ‘17 – 9 =?’ can be read in several ways: ‘17 minus 9 equals what?’ ‘17 subtract 9 equals what?’ or ’17 take away 9 equals what?’ All of these linguistic expressions create a conceptual model in the mind. You begin with 17 objects and then you take away/subtract/minus 9 objects from these. The conceptual model and the language container formed by this language are inadequate for later instances of subtraction. For example, -32 – (-7) does not lend itself to this model. To avoid these and related problems, students should learn to express mathematical ideas in multiple linguistic forms. For example, the expression ‘17 – 9 =?’ should also be expressed as:  17 minus/subtract/take away/deduct 9 equals what?  What do you get when subtract/deduct/minus/take away 9 from 17 gives you what?  What is the difference of 17 and 9?  How much more is 17 than 9?  How much less is 9 than 17? Etc. A multiplicity of linguistic expressions makes students more flexible in the use of words, voice, and order.

E. Developing Translation Ability: Solving Word Problems One of the difficulties children face is to answer a problem posed in words even when they can do the relevant calculation. This is evident on local, state, national and international assessments.

9 Because of the correlational nature, there has always been considerable interest in the relationship of reading and mathematics word problem solving. Educators have focused on developing vocabulary and strengthening reading comprehension to improve problem solving. This usually means learning and identifying key phrases and the corresponding arithmetical operations. However, this does not help the child to make deeper and lasting connections between language and problem solving. Word problems are, on the surface, computations in linguistic form. In truth, they are examples of integrating cognitive, linguistic, and mathematical thinking. However, most children and adults dread word problems as they view them as difficult and unnecessary. They perform poorly on word problems. Solving word problems involves several language and mathematics skills: . Recognizing embedded mathematical vocabulary. Understand and reformulate the problem in one’s own words. (What is given?) . Understanding the mathematical concepts and operations behind those words. (What is being asked? What needs to be done?) . Translating English into mathematical expressions (Forming equations, strategy seeking, and procedures) and the calculated values in terms of what was asked in the problem. (Is this what we needed to do?) The ability to solve word problems, therefore, lies in proficiency in translation. The complexity of these translations suggests why children find word problems difficult. Facility in translation, in turn, is dependent on the development of mathematical language. A language expression in mathematics suggests the related conceptual schema. Let us consider simple subtraction: (a) David has 14 books and John takes away 9. How many are left? (b) David is 6 feet tall and John is 4 feet tall. How much taller is David than John? (c) School is 16 blocks from David’s house. David has already walked 7 blocks. How many-more blocks must he walk? Each one of these three linguistic models suggest a conceptual schema, instructional material and teaching strategy: (a) take away suggests counters and ‘counting on or down’ strategy; (b) how much more suggests Cuisenaire rods and measurement; (c) how many more suggests number line and comparison/missing-addend. Limited language and conceptual frameworks do not prepare children well for problem solving. Children presented with word problems requiring identical operations on identical numbers but changing the wording and specific problem contexts exhibit differences in outcomes. The language variables that contribute to problem difficulty are: familiarity of the situation described in the problem; the arrangement of problem; the number of unfamiliar objects, and nonessential elements; whether the story problem were intrinsically interesting; how difficult the vocabulary was; whether the problems were presented as declarative sentences or as questions; problem statement; placement of the question; and so forth.

10 F. The Role of Questions in Acquiring Mathematics Language Language acquisition is a function of interaction with others. The teacher can facilitate this interaction by asking many hypothetical questions. The questioning process helps children to understand ideas and their correct use. . Questions instigate language; . Language instigates models; . Models instigate thinking; . Thinking creates understanding; . Understanding produces competent performance; . Competent performance develops self-esteem; and . Self-esteem is the motivating force for all learning. Teachers should ask the full range of questions—from recall to forming conjectures, from convergent to divergent thinking, to get complex answers and more language. The nature and type of question should reflect the content to be developed. . How many10s in one hundred? . What number occurs next in the sequence 1, 3, 6, 10, …? . How could you test a number to see if it is divisible by 6? . Give a three-digit number that is divisible by 2, 3, 5 and 10. . Why is the sum of two odd numbers always even?

G. Mathematical Instructions Each set of mathematics problems has instructions, which demand the performance of operation(s) or action(s). Before students can solve the problem, they must understand the instruction(s). When demands are explicit, the student knows what to do, but when they are not, the student may be at a loss. Though the reading level of the instructions and the problem may be at a lower level, students, particularly those with reading problems, may find them difficult understand. Students can solve a problem, if they know what is being asked and are familiar with the content. Two types of instructions are used: explicit or implicit. In explicit instructions, the student is told clearly what to do. When the instructions are explicit, students know what to do if they know the content. For example, (a) multiply the numbers 1.2 and 1.3, (b) find the product of 2a and 3ab, or (c) differentiate the function 2x3sin(5x) by parts. Instructions are clear without knowing the content. If students know multiplying decimals, finding the product’ or the process of differentiating by parts, they can solve the problems respectively. In these, the success was dependent on knowing the content. Instructions were not the hindrance.

11 Many students have difficulty understanding the instructions because they do not know the content/procedure. Knowing the meaning of words involved in the problem is not enough. For example, (a) Using long division procedure, find the quotient of 7.25 divided by .025, (b) Find the ratio of 3½ and 2¾, or (c) Find the greatest common factor of numbers 6, 8, and 54. Students may know what long division, ratio, and greatest common factor mean. However, students cannot solve the problems, as they may not know the process of long division, etc. In implicit instructions, the information is assumed, indirectly given, embedded in the problem, or students need to supply it to solve the problem. For example, . What is the measurement of the smallest angle of the triangle, if the two angles of the triangle are 70 and 80? . John practices 1.2 hours each day. He needs to practice 20 hours before the next concert. How many days of practice does he need? Here, the instructions are implicit. Students who have had practice with these types of problems are more likely to answer them correctly. I asked a student to read the following algebraic expression: Evaluate: 4p + 2q2 where p = 3 and q = 7 “What does the word ‘evaluate’ mean?” “I do not know.” “Have you ever seen this word before?” “Maybe?” How can the student be expected to solve these types of problems if the instructions are not clear to them? Understanding instructions is dependent on mastery of language of mathematics. Teachers must help students to understand instructions. They should help the student to create a set of questions to decode instructions. What are the instructions in this problem? What are they asking me to do? What does the word “solve” mean? What does the word “evaluate” mean? What does the word “determine” mean? Many students do not read the instructions or read but do not understand them. They determine their task based on the context and experience. Teachers need to teach how to read instructions and translate information. This means that teachers should emphasize vocabulary, syntax, translation of mathematical terms into symbols and from mathematics into English and mathematical ways of thinking (reasoning, critical thinking and communication). 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

12 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 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 work: 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

13 activity using pictures and symbols. The teacher or a student can also act the word out.  Writing 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.  Oral 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 • 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

14 . 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. H. Teaching Mathematics Language To improve mathematics proficiency for all students, children and adults, I advocate second language learning techniques. Mastery in the native language can be used as a ‘learning to learn platform’ to transfer fluency and conceptual schemas from the native language to the language of mathematics. Ability to transfer assumes fluency and presence of mathematics conceptual schemas in the first language. If children do not have mathematics conceptual schemas in their native language then we should develop mathematics conceptual schemas in the second language. It is counter productive to develop schemas in the native language and then transfer them to the second language. For developing a mathematics language, we create an environment that is rich in language and mathematics. The aim is to develop language in the context of specific mathematics content. Appropriate concrete models are used to model the concepts related to that mathematics content. Children are encouraged to talk about their learning. They record their activity pictorially and symbolically. Teachers introduce words and language containers using these representations. Students replicate activities through concrete models, representations, and symbols. Students are taught and encouraged to translate English expressions involving data into mathematics language and vice versa. First, students are given mathematical expressions to translate into number stories. For example: Given 2x + 2y = 48, was translated into: . Twice the length and twice the width, of a rectangle, is equal to forty- eight. . Twice John’s age added to twice Mary’s age is equal to forty-eight. The mathematical expression ‘7 + 4 =?’ was translated into:

15 . My friend shot 7 monsters in the video game on his first turn and 4 more on his second turn. How many monsters did he shoot? Later, stories with data are given and students write the mathematics expression. . Mary has 22 apples and John has 13 bananas. How many more pieces of fruits does Mary have than John? This was translated into: 22 -13 =? . The sum of two numbers is 3 times their difference. This was translated into: (x + y) = 3 (x – y) In an environment rich with opportunities of communicating mathematics ideas, students notice mathematics around them, use it, and initiate mathematical behaviors – more symbolization, more mathematics talk and writing. Students learn to translate from concrete models and the native language to the mathematics language and vice versa. Students with training in writing mathematics stories exhibit a better understanding and are less afraid of word problems. They have a better command of mathematics and are able to identify the correct operations. Development of an appropriate mathematics language should be a major emphasis in teacher training programs. Unless future teachers are encouraged to develop a rich understanding of the language of mathematics, the reform effort will be ineffective.

Bibliography: Allardice, B. (1997). The development of Written representations for some mathematical concepts, Journal of Children’s Mathematical Behavior, 1(4). Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S.B. (1999). Children’s Mathematics: Cognitively Guided Instruction. Portsmouth, NH: Heinemann. Cook, V. (2001) Second Language Learning and Language Teaching, London: Arnold, Hodder Headline Group Focus on Learning Problems in Mathematics. (2001). Language Issues in the Learning of Mathematics, volume 23(2&3). Framingham, MA: CTLM Ginsberg, H. (1989), Children’s Arithmetic, Austin, TX: Pro-ed Hughes, M. (1986), Children and numbers: Difficulties in Learning Mathematics, Oxford: Blackwell Kedm, G. (1984), The classifications of Problem Solving research variables, In G.A. Goldin & C.E. McClintock (Eds.), Task Variables in Mathematics Problem Solving, (1-21), Philadelphia: The Franklin Press Kouba, V.L., Brown, CA. Carpenter, T.P., Lindquist, M.M., Silver, E.A. & Swafford, J.O. (1998), Results of the Fourth NAEP Assessment of Mathematics & Number, Operations and Word Problems, Arithmetic Teacher, 35(8):14-19 National Council of Teachers of Mathematics. (2000). Principles and Standards of School Mathematics. Reston, VA: Author Pimm, D (1981), ‘Mathematics? I speak it fluently’, In A. Floyd (ed.) Developing Mathematical Thinking, Wokingham: Addison Wesley. Sharma, M.C. (2005). Mathematics as a Second Language: Reading, Writing, and Doing Mathematics. Framingham, MA: CTLM Worthington, M. and Carruthers, E. (2003), Children’s Mathematics: Making Marks, Making Meaning, London: Paul Chapman Publishing

16 Mahesh Sharma Mathematics for All Center for Teaching/learning of Mathematics Framingham and Wellesley, MA 508 494 4608 [email protected] www.mathematicsforall.org Blog: www.mathlanguage.wordpress.com

Professor Mahesh Sharma is the founder and President of the Center for Teaching/Learning of Mathematics, Inc. of Wellesley and Framingham, Massachusetts and Berkshire Mathematics in England. Berkshire Mathematics facilitates his work in the UK and Europe. He is the former President and Professor of Mathematics Education at Cambridge College. Where he taught mathematics and mathematics education for more than thirty-five years to undergraduate and graduate students. Internationally known for his groundbreaking work in mathematics learning problems and education, particularly dyscalculia and other specific learning disabilities in mathematics. He is an author, teacher and teacher-trainer, researcher, consultant to public and private schools, as well as a public lecturer. He has been the Chief Editor and Publisher of Focus on Learning Problems in Mathematics, an international, interdisciplinary research mathematics journal with readership in more than 90 countries, and the Editor of The Math Notebook, a practical source of information for parents and teachers devoted to improving teaching and learning for all children. He provides direct services of evaluation and tutoring for students (children as well as adults) who have learning disabilities such as dyscalculia or face difficulties in learning mathematics. Professor Sharma works with teachers and school administrators to design strategies to improve mathematics curriculum and instruction for all. Contact Information: Mahesh C. Sharma [email protected] 508 877 4089 (H) 508 494 4608 (C) 508 788 3600 (F)

17 Center for Teaching/Learning of Mathematics Programs and Services CT/LM offers programs and materials to assist teachers, parents, therapists, and diagnosticians to help children and adults with their learning difficulties in mathematics. We conduct regular workshops, seminars, and lectures on topics such as: 1. How does one learn mathematics? This workshop focuses on psychology and processes of learning mathematics—concepts, skills, and procedures. The role of factors such as: Cognitive development, language, mathematics learning personality, pre-requisite skills, conceptual models, and how key developmental milestones in mathematics learning such as: number conceptualization, place value, fractions, integers, algebraic thinking, and spatial sense are achieved. Participants learn strategies to teach their students more effectively. 2. What are the nature and causes of learning problems in mathematics? This workshop focuses on understanding the nature and causes of learning problems in mathematics. We examine existing research on diagnosis, remedial and instructional techniques in dealing with these problems. Participants become familiar with diagnostic and assessment instruments for learning problems in mathematics. They learn strategies for working more effectively with children and adults with learning problems in mathematics. 3. Content workshops. These workshops are focused on teaching key mathematics milestone concepts and procedures. For example, they address questions such as: How to teach arithmetic facts easily and effectively? How to teach fractions more effectively? How to develop the concepts of algebra easily? In these workshops, we use a new approach called Vertical Acceleration. In this approach, we begin with a very simple concept from arithmetic and take it to the algebraic level. 4. What to look for in a results-oriented mathematics classroom: This is a workshop for administrators and teachers to understand the key elements necessary for an effective mathematics classroom. We offer individual diagnosis and tutoring services for children and adults to help them with their mathematics learning difficulties and learning problems, in general, and dyscalculia, in particular. We provide: 1. Consultation with and training for parents and teachers to help their children cope with and overcome their anxieties and difficulties in learning mathematics, including dyscalculia. 2. Consultation services to schools and individual classroom teachers to help them evaluate their mathematics programs and teaching and help design new programs or supplement existing ones in order to minimize the incidence of learning problems in mathematics. 3. Assistance for the adult student who is returning to college and has anxiety about his/her mathematics. 4. Assistance in test preparation (SSAT, SAT, GRE, GMAT, MCAS, etc.) 5. Extensive array of mathematics publications to help teachers and parents to understand how children learn mathematics, why learning problems occur and how to help them learn mathematics better. The Math Notebook (TMN) Articles in TMN address issues related to mathematics learning problems, diagnosis, remediation, and techniques for improving mathematics instruction. They translate research into practical and

18 workable strategies geared towards the classroom teacher, parents and special needs teachers/tutors. Topics covered range from K through College mathematics instruction. Selected Back Issues of The Math Notebook:  The Problem of Missing Addend (single)  Children’s Understanding of the Concept of Proportion – Part 1 and 2 (double)  A Topical Disease in Mathematics: Mathophobia (single)  Pattern Recognition and Its Application to Math (double)  Mathematics Problems of the Junior and Senior High School Students (double)  Mathematically Gifted and Talented Students (double)  Types of Math Anxiety (double)  Developing Visual Thinking in Children (double)  Memory and Mathematics Learning (double)  Problems in Algebra - Part 1 and Part 2 (special)  Games and Their Uses in Mathematics Learning (double)  Reversal Problems in Mathematics and Their Remediation (double)  How to Take a Child From Concrete to Abstract (double)  Levels of Knowing Mathematics (double)  Division: How to Teach It (double)  Soroban: Instruction Through Concrete Learning (double)  Mathematics Culture (double)  Mathematics Learning Personality (double)  Common Causes of Math Anxiety and Some Instructional Strategies (double)  On Training Teachers and Teaching Math (double)  Will the Newest “New Math” Get Johnny’s Scores Up? (double)  Concept of Number I and II (special)  Dyslexia, Dyscalculia and Some Remedial Perspectives For Mathematics Learning Problems (special)  Place Value Concept: How Children Learn It and How To Teach It (special)  Cuisenaire Rods and Mathematics Teaching (special)  Authentic Assessment in Mathematics (special) FOCUS on Learning Problems in Mathematics FOCUS, an interdisciplinary journal, for the last thirty years, has made available the current research, methods of identification, diagnosis and remediation of learning problems in mathematics. It published original articles from fields of education, psychology, mathematics, and medicine having the potential for impact on classroom or clinical practice. Topics included reports of research on processes, techniques, tools and procedures useful for addressing problems in mathematics teaching and learning: descriptions of methodologies for conducting, and reporting and interpreting the results of various types of research, research-based discussions of promising techniques or novel programs; and scholarly works such as literature-reviews, philosophical statement or critiques. Focus has made real contribution in the field of mathematics education, learning problems in mathematics and how to help children and adults in dealing with their mathematics difficulties. Selected back issues of Focus: Volume 3, Numbers 2 & 3 (1981): Educational Psychology and Mathematical Knowledge Volume 4, Numbers 3 & 4 (1982): Fingermath: Pedagogical Implications for Classroom Use

19 Volume 5, Number 2 (1983): Remedial and Instructional Prescriptions for the Learning Disabled Student in Mathematics Volume 5, Numbers 3 & 4 (1983): Mathematics Learning Problems and Difficulties of the Post Secondary Students Volume 6, Number 3 (1984): Education of Mathematically Gifted and Talented Children Volume 6, Number 4 (1984): Brain, Mathematics and Learning Disability Volume 7, Number 1 (1985): Learning Achievement: Implications for Mathematics and Learning Disability Volume 7, Numbers 3 & 4 (1985): Using Errors as Springboards for the Learning of Mathematics Volume 8, Numbers 3 & 4 (1986): Dyscalculia Volume 9, Numbers 1 & 2 (1987): Computers, Diagnosis and Teaching (Part 1&2) Volume 11, Numbers 1 & 2 (1989): Visualization and Mathematics Education Volume 11, 3 (1989): Research on Children’s Conceptions of Fractions Volume 12, Numbers 3 & 4 (1990): What Can Mathematics Educators Learn from Second Language Instruction? Volume 13, Number 1 (1991): Students’ Understanding of the Relationship between Fractions and Decimals Volume 14, Number 1 (1992): The Psychological Analysis of Multiple Procedures Volume 15, Numbers 2 & 3 (1993): Vygotskian Psychology and Mathematics Education Volume 17, Number 2 (1995): Perspective on Mathematics for Students with Disabilities Volume 18, Numbers 1-3 (1996): Gender and Mathematics: Multiple Voices Volume 18, Number 4 (1996): The Challenge of Russian Mathematics Education: Does It Still Exist? Volume 19, Number 1 (1997): Components of Imagery and Mathematical Understanding Volume 19, Number 2 (1997): Problem-Solution Relationship Instruction: A Method for Enhancing Students’ Comprehension of Word Problems Volume 19, Number 3 (1997): Clinical Assessment in Mathematics: Learning the Craft Volume 20, Numbers 2 & 3 (1998): Elements of Geometry in the Learning of Mathematics Volume 22, Numbers 3 & 4 (2000): Using Technology for the Teaching and Learning of Mathematics Volume 23, Numbers 2 & 3 (2001): Language Issues in the Learning of Mathematics Volume 28, Number 3 & 4 (2006): Concept Mapping in Mathematics

Index of articles in Focus on Learning Problems in Mathematics from Volume 1 to 30 available on request. Individual issue $ 10.00 Double issue $ 15.00 Each Volume (four issues) is available separately $ 25.00 Whole set of 30 volumes is available for $250.00 Back Issues of TMN: Single issue ($3.00); Double issue ($5.00); Special issue ($8.00) (Postage and handling 20% of the total extra) to order send email: [email protected] or Fax: 508 788 3600.

20 Other Publications Dyslexia and Mathematics Language Difficulties $15.00 How to Master Arithmetic Facts Easily and Effectively $15.00 Guide for an Effective Mathematics Lesson $15.00 How to Teach Fractions Effectively $15.00 Math Education at Its Best: Potsdam Model $15.00 How to Teach Number to Young Children $15.00 Dyscalculia $15.00 The Games and Their Uses in Mathematics Learning $15.00 How to Teach Subtraction Effectively $12.00 Literacy & Numeracy: Comprehension & Understanding $12.00 The Questioning Process: The Basis of Effective Teaching $12.00 Visual Cluster Cards $12.00 Videos How Children Learn: Numeracy $30.00 (One-hour long video interviewing Professor Mahesh Sharma on his ideas about how children learn mathematics) How To Teach Place Value $30.00 (Strategies to teach place value effectively) Numeracy Videos (Complete set of six for $150.00 and individual for $30.00) 1. Teaching arithmetic facts, 2. Teaching place value, 3. Teaching multiplication, 4. Teaching fractions, 5. Teaching decimals and percents, and 6. Professional development: Teachers’ questions) Most children have difficulty in mathematics when they have not mastered the key mathematics milestones in mathematics. The key milestones for elementary grades are: Number conceptualization and arithmetic facts (addition and multiplication), place value, fractions and its correlates—decimal, percent, ratio and proportion. These videos and DVDs present strategies for teaching these key mathematics milestone concepts. They apply Prof. Sharma’s approach to teaching numeracy. These were videotaped in actual classrooms in the UK. P l e a s e s e n d o r d e r t o ( A d d 2 0 % e x t r a f o r p o s t a g e a n d h a n d l i n g ) : CENTER FOR TEACHING/LEARNING OF MATHEMATICS 754 Old Connecticut Path, Framingham, MA 01701 508 877 4089 (T) 508 788 3600 (F)

21 [email protected] www.mathematicsforall.org