sexta-feira, 31 de agosto de 2012

English Language Learners in Math

English language learners have unique needs when it comes to mathematics instruction. Find out about several key strategies to help ELLs succeed.

For some students, math seems to be a foreign language, consisting of words and concepts that don't mesh with their everyday experiences. For these students, successful teachers find ways to make math understandable, relevant, and familiar. In some ways, it is all about finding a common language.
So what is a teacher to do when there truly is not a common language shared between teacher and student?

Math classes for English Language Learners (ELLs) can be especially challenging because students are faced with learning both mathematics and English at the same time. Math teachers can and must make every effort to reach out to these students to create a class that is both positive and rewarding.

By employing the following strategies and techniques, you can help your ELL students increase their knowledge and understanding of math topics.

Use Small Groups

The use of pairs or small groups is an instructional strategy that can be very effective for ELL students.
By grouping students, you can:
·          encourage communication and interaction in a non-threatening and more relaxed setting.
·          help students feel more comfortable to ask questions or seek explanations.
·          promote a positive support system for your ELL students.
·          manage large classes with diverse student needs more effectively.

Vary Math Instruction and Provide Interesting Problems

As a math teacher of ELL students, it is important to utilize multiple instructional approaches and to consider the individual learning styles of your students. This will help you better meet the varied needs of your students.
Try one or more of the following instructional strategies.
·        Manipulatives and models. Where appropriate, use models or manipulatives to demonstrate concepts and/or processes. Allow ELL students to use them as well to demonstrate their learning. 
·        Think-Alouds. Use a "think-aloud" technique to narrate the problem-solving process (including algorithms). 
·        Informal language. To increase understanding, use informal language as you demonstrate the various thought processes and steps to follow in solving a problem. 
·        Clarity checks. Be sure to check for understanding of the task and processes involved before students get started working on the assignment. ELL students often do not seek clarification for fear of calling attention to themselves. 
·        Context. Present activities that involve application problems in contextualized situations. These activities should encourage critical thinking and reasoning along with basic skills development and practice. Engaging your ELL students in this way is important because it makes learning relevant to the real-life experiences of your students. Sports, entertainment, and games can be good themes to do this. 
·        Diagrams and Graphs. Encourage the use of diagrams and other visual aids to help your ELL students develop concepts and understanding. For written and verbal communication, increase your focus on reasoning and decrease the focus on language. This emphasis can help to encourage your ELL students to expand their mathematical abilities without getting bogged down with issues related to language acquisition. 
·        Assignments. Consider limiting the number of problems you assign to ELLs to avoid overwhelming them. Focus on fewer problems that get at essential concepts and skills.

Teach Math Vocabulary

Math classes can present extra challenges for ELL students because they must learn the specific content vocabulary and expressions, along with their second language acquisition. Help your ELL students by directly teaching math vocabulary. Use the following ideas to plan your vocabulary building activities.

·        Topical Terms. Identify and teach specific vocabulary that must be understood for each assigned activity. 
·        Common Math Terms. Teach the meanings of common math terms that have other definitions outside of the context of math-for example root, face, mean, and prime. 
·        Solutions. Create a plan for how you will help ELL students acquire the language of mathematics. 
·        Visual Aids. Consider using visual aids, multiple examples, and student explanations as possible techniques to help your ELL students grasp unfamiliar math terms. 
·        Note Cards. Encourage your ELL students to keep note cards to record math terms and vocabulary in their own words, sometimes with the use of their native language. Remind your ELL students to reference these note cards and to add to them as their understanding grows. 
·        Math Journals. Use journals to practice and strengthen new language skills and math terms in a non-threatening manner. Some ELL students may feel comfortable using their native language in their journals as a way to help solidify their understanding of math concepts. 

Monitor Your Interactions with ELL Students

To help your ELL students follow lectures and understand class discussions, you will need to be aware of your speech and consider simplifying it when you can. Some ways to do this include:

Effective Communication Strategies for Teachers
·        Pause frequently.
·        Paraphrase often.
·        Emphasize key ideas and vocabulary through intonation.
·        Write key terms and concepts on the board.
·        Use pronouns clearly.
·        Shorten sentences.
·        Increase wait time for students to answer and process information.

Use Prior Learning as a Starting Point

It is essential that you take into account the ELLs unique experiences, prior learning, and individual strengths to develop appropriate instructional strategies. Other cultures have different approaches to mathematics that even the ELL may be unaware of.
Mathematical Concepts That May Differ or Be Difficult
Measurement may be especially challenging for ELL students, as their prior instruction most likely covered the metric system.

Fractions may be unfamiliar to ELLs. Some ELL students may have come from an educational environment where decimals received more emphasis than fractions.

The discipline of Geometry in particular has many terms that may cause difficulties in understanding.

In some cases, algorithms may have been learned differently. Some ELL students may be used to algorithms that are different from traditional algorithms taught in your curriculum. Allow students the opportunity to share their algorithms. Use this as a learning opportunity by comparing algorithms and analyzing similarities and differences.

Value Student Backgrounds to Improve Performance

Be aware of and draw from your ELL students' prior knowledge.
You should:

·        Make sure ELL students know that their experiences and culture are valued. This will help their attitudes and increase their motivation.
·        Focus on meaning. When your ELL students share answers and present ideas, remember to focus on the meaning they are conveying, not on their grammar and usage.
·        Be flexible with student use of native language. You may want to have your ELL students record answers or solution steps in their own native language. You can have their work translated, if desired. This will help ELL students focus on the concepts and reasoning involved, without being slowed or hindered by their developing language skills.

Multi-Stepped Problem Lesson Idea

Present a multi-stepped problem to your class. Have your students identify the steps that they would need to undertake to solve the problem, but just identify the steps; do not have your students solve the problem. This will help ELL students practice their language skills by reading and interpreting problems. This type of exercise will help students practice language skills without worrying about solving the problem. This exercise can be used frequently as a quick warm-up activity.

Success Is Possible

As a concerned math teacher, your instructional goal should be to facilitate full participation in all aspects of a rich mathematical educational experience for all students. You can do this by lessening or removing some of the barriers to learning that are connected to the process of second language acquisition. Keeping a few of these instructional strategies and techniques in mind can help you make your math instruction better for not just ELL students, but for all of your students.

quinta-feira, 30 de agosto de 2012

How much input do you need to speak English fluently?

by Tomasz P. Szynalski

How much information do you need?

Few people realize that learning a language fluently is a much more memory-intensive task than, say, learning organic chemistry or the history of Europe at an expert level.

Let's consider the number of facts you need to know to produce correct English sentences with ease. Certainly you must know the meanings and pronunciations of something like 10,000 words and phrases — the contents of a medium-sized dictionary. But this is only half the picture. The other half are thousands upon thousands of little facts which tell you when to use different words and how to combine them with other words:
·         We say “I walk”, but not “he walk”; “he says” but not “he mays”; “Is she young?”, but not “Looks she young?”; “He did it”, but not “He didn’t it”; “She looked beautiful”, but not “She dressed beautiful”.
·         We say “Mary likes cheese very much”, but not “Mary very much likes cheese”; “He might have eaten the cake”, not “He could eat the cake” or “He has might eaten the cake”; “What did he eat?”, not “What he ate?”, “What he did eat?” or “He ate what?”.
·         We do an exercise, but make a mistakemake a phone call, but have a conversationdo a job, but take a breaktake a step, but make a jump.
·         You can have a bad/terrible headache, but not astrong/heavy headache; you can getgreat/enormous satisfaction, but not big satisfaction; you can be a heavy smoker, but not a hard/strong smoker; you can have a heated debate, but not a burning debate; you can have afast car, but not a fast look; you can clean your teeth, but you cannot clean the dishes.
·         We talk about somethingcomment onsomething and discuss somethingsucceed insomething, but fail at somethingask a questionof somebody, but have a question for somebody;accuse somebody of something, but blame somebody for somethinganswer an e-mail, butreply to an e-mail.
·         You can give an opinion, but not an advicebuy a cake, but not a breadmove a table, but not a furnitureshare a fact, but not an information.
·         You can ask somebody to do something, but notsuggest somebody to do something; you can tell somebody something, but not explain somebody something; we encourage somebody to dosomething, but discourage somebody from doing something; we tell/want/get/allow somebody to go, but let/make/see/hear somebody go.
A book like Michael Swan’s Practical English Usage has 600 pages of facts like these. And I wish I could tell you that they are unnecessary — that you can just ignore them — but the fact is, Iobey virtually all of these rules when I speak or write in English, and so does anyone who is fluent.
If knowing a language requires so much knowledge, then how come everyone can speak at least one language fluently? We are not all Einsteins. There are many native English speakers who are not very skilled at acquiringknowledge, yet all of them successfully use the English that is spoken in their community (whether it is standard English or the Black English Vernacular).
A face you recognize.
The reason why we can memorize a huge database of language facts is the same reason why we can recognize faces. Consider how much information is required to recognize that Bill is really Bill: the dimensions of the head, the colors, shapes, relative sizes, and positions of the eyes, nose, eyebrows, lips, teeth, ears, chin, cheeks, forehead, hairline, hair, wrinkles, spots and facial hair. If I were to write down a precise description of just one face, imagine how many sheets of paper that would take, and how hard it would be to memorize. Yet all of us (even people with a poor memory) can recognize thousands of faces and it takes only a blink of an eye in each case.
We can do this because we don’t have to memorize people’s facial features like we would memorize history facts. Our brain has a special module which can instantly grab all the data as we look at a face. Then, when we look at a face, this module can answer the question “Do I know this face?”. It all happens effortlessly and subconsciously. We never have to think “Um, the convex shape of this person’s nose, the distance between the eyes, and the asymmetrical upper lip match my friend Peter”.
Just as everyone (smart or not) has a face recognition module, everyone has a language module. This module stores facts about word meaning, word usage, grammatical structures, pronunciation, etc., so you don’t have to memorize them like you would memorize history facts. While the face recognition module lets you answer the question “Do I know this face?”, the job of the language module is to produce correct sentences based on what it has learned about language.
How does the language module get its facts about language? Unfortunately, it takes more than a quick look. Long lists of rules (like the one above) won’t help either — the language module evolvedlong before there were grammar books, so it doesn’t understand grammar rules. The module get its information from example sentences. As you read and listen to correct sentences in a language, it builds, piece by piece, a database of facts about that language.

How much input do you need?

All right, so the language module in your brain needs correct sentences. But how many sentences do you actually need to become fluent in a language?
First of all, the question is a bit misleading, because there isn’t a single answer for all situations. The number of sentences you need will depend on many factors:
·         the difficulty of the sentences (e.g. if you get too easy or too difficult sentences, you won’t learn much)
·         the style of the sentences (if you read too much literary language, it will not help you speak)
·         your pace (if you get more sentences per day, you need fewer sentences in total, because you forget less information)
·         how you get the sentences and how much attention you pay to them (when reading, it is possible to analyze each sentence much more carefully, so you can get more information out of each sentence, but you also read more slowly)
·         your innate skills (some people need more input before they can speak, others “get it” very quickly)
·         how close your first language is to the language you are learning (a speaker of Dutch needs much less input to learn English than a speaker of Japanese)
How much input did I get? It took me about 3 years to get from basic English skills to fluency. During those 3 years, I was exposed to about1,000,000 English sentences (not necessarily different sentences). About 400,000 of these were written sentences (books, SRS reviews, dictionaries, classroom reading); 600,000 were spoken sentences (TV, recordings, listening to teachers, listening to my American cousin, classroom listening).
Note that these are very rough estimates. Theactual number of sentences that I got during that 3-year period may well have been 700,000 or 1,500,000.

“Holy moly!”

I know. One million is a big number. But when youbreak it down, it looks far less scary:
1,000,000 sentences over 3 years equals roughly 6,400 sentences per week - for example, 1,600 written sentences per week (roughly 60 pages in a typical book) plus 4,800 spoken sentences per week (roughly 6 hours of audio (non-stop talking))
So if you want to follow in my footsteps, you’ll have to get about 60 pages of written English and6 hours of spoken English per week — for three years. (I am assuming you already have some basic English skills that enable you to understand this article. If you are a total beginner, you will have to get to that level first.) If you think 60 pages and 6 hours is a lot, consider the following points:
·         Using an English-English dictionary with example sentences and SRS reviews take care of perhaps 15 pages of written English per week. This leaves 45 pages per week for traditional reading (websites, books).
·         Reading 45 pages per week may seem scary when you are just beginning to read in English. But I promise you — you will be devouringEnglish texts in no time!
·         Remember that things like listening to your teacher, having conversations in English, watching videos on YouTube, watching House M.D., playing Mass Effect, etc. all count as “listening time”. (This does not mean that 6 hours of playing a videogame fulfills the 6-hour requirement — you need 6 hours of talking without breaks, as in an interview.)

quarta-feira, 29 de agosto de 2012

Teaching Mathematics and English to English Language Learners Simultaneously - PART II

Mathematical vocabulary

The math register includes a variety of words, phrases, and expressions, which can be placed into four categories (Figure 1). High-frequency vocabulary, learned in almost any setting, overlaps social and academic language and consists of words and phrases most commonly heard and used. General vocabulary, primarily learned in classrooms and more formal settings, allows students to communicate about specific topics.

Figure 1
Vocabulary types common in mathematics classrooms (adapted from Ernst-Slavit and Slavit, 2007)

High-frequency vocabulary
Mostly social language; Terms used regularly in everyday situations
small, orange, clock
General vocabulary
Mostly academic language; Terms used
in school but not directly associated
with mathematics
combine, describe, consequently
Specialized vocabulary
Academic language; Terms broadly associated with mathematics
number, angle, equation, average
Technical vocabulary
Academic language; Terms associated with a specific mathematical topic
perfect numbers, supplementary angles, quadratic equations, cosine, mode

In addition to these more general kinds of words and phrases, mathematics teachers must also develop specialized and technical vocabulary, which are words and phrases specific to the mathematical content under discussion. Wong-Fillmore & Snow (2000) have listed a series of words that pose many challenges for ELLs, such as terms that express various kinds of quantitative relationships as well as everyday words that provide logical links in sentences typical to mathematical word problems (Figure 2).
Figure 2
Problematic words for ELLs commonly used in mathematics textbooks and classrooms.

Words that express quantitative relationships
Words that link phrases and sentences and express a logical relationship
hardly, scarcely, rarely, next, last, most, many, less, longer, older, younger, least, higher
if, because, unless, alike, same, different from, opposite of, whether, since, unless, almost, probably, exactly, not quite, always, never

Representing information in non-linguistic ways is also an important consideration when "talking math." For example, the idea of slope can be expressed using graphs of lines, algebraic symbols and formulas, tables of values, or with contextual information (e.g., the fixed cost of an item is the slope of a cost function for that specific item). Further, there are a variety of linguistic expressions commonly used to refer to the general concept of slope, including "rate of increase/decrease/change," "linear change," "degree of inclination," and "rise over run." Students must draw from all four of the vocabulary types when participating in mathematical conversations of this kind. As all teachers of mathematics know, specific language considerations are also needed due to the precise meaning of mathematical terms; for example, slope is a "rate of change," but not all rates of change have a slope. Hence, ELL students need to be made especially aware when their language can be "loose," and when it must be precise.

Mathematical grammar

At the sentence level, there are language patterns and grammatical structures specific to mathematics. These include the use of logical connectors (e.g., "consequently," "however") that in regular usage signal a logical relationship between parts of a text, but in mathematics signal similarity or contradiction. Likewise, the use of comparative structures (e.g., "greater than" and "less than," "n times as much as") and prepositions (e.g., "divided by," "divided into") pose serious difficulties for students who are trying to learn the content while, at the same time, trying to learn the language used to access that content. Semantic aspects of language can also pose difficulties, as in the following example (Dale & Cuevas, 1992): Three times a number is 2 more than 2 times the number. Find the numbers.

Solving this problem requires a recognition of how many numbers are involved, the relationships between them, and which ones need to be identified. In addition, ELLs (Dale and Cuevas, 1992) and other students (Clement, 1982) often encounter difficulties when they attempt to read and write mathematical sentences in the same way they read and write narrative text. That is, students may try to literally translate a mathematical concept expressed in words into a concept expressed in symbols. For example, the algebraic phrase "the number a is five less than the number b" is often translated into a=5-b, when it should be a=b-5.

Teaching strategies for learning and talking mathematics

The above discussion was intended to provide information to assist teachers to better understand the language abilities and needs of their learners. But we still must ask, "How do mathematics teachers teach their students mathematical thinking if their students speak very little English?" Or, as we are often asked, "How do I reach my ELLs?" Although there is no simple answer for this question, the truth is that in many mathematics classrooms, teachers are using a variety of instructional strategies that have proven useful for reaching all students, but, in particular, those who are learning English as a second language (see Figure 3 for some general approaches currently in use). It is important to remember the demands being placed on ELLs in these learning situations:

Figure 3

Program models and approaches for teaching mathematics (and other content) to English language learners.

Content-based language instruction
(TESOL, 2006)
Teachers use and adapt materials from the mathematics curricula as a vehicle for developing language and content.
Also called integrated language and content instruction, it is usually taught by a specialist in both mathematics and ESL.
Sheltered instruction
(TESOL, 2006)
The mathematics curricula is adapted to accommodate students’ level of English language proficiency.
Teachers use mathematical materials that are challenging, but may be at a lower reading level. Abstract concepts are broken down to concrete attributes, and vocabulary skills are enhanced. It serves to transition students from the ESL class to the academic mainstream.
Specially-Designed Academic
Instruction in English (SDAIE)
(Diaz-Rico & Weed, 2006)
Teaching of grade-level subject matter in English while also promoting English language development.
Teachers are encouraged to focus on
(a) their own speech, by limiting the use of idioms, speaking slowly, and using everyday meaningful vocabulary; (b) the use of visuals and contextual clues, including gestures to convey meaning; and, (c) lesson planning that uses and builds on students’ background knowledge.
Guided Language Acquisition Design (GLAD)
Actively using language across content areas.
Lessons are planned around
(a) an engaging topic; (b) motivation; (c) multiple forms of review and evaluation; and (d) specific vocabulary, concepts, skills, and higher order thinking skills.
Optimal Learning Environment (OLE)
(Ruiz & Figueroa, 1995)
Outlines specific conditions that promote student learning of content.
These include high expectations, immediate feedback, building community, placing meaning before form, and immersing students in print.
The Sheltered Instruction
Observational Protocol (SIOP)

(Echevarria, Vogt, & Short, 1999)
Promotes self-evaluation and reflection.
The protocol provides extensive criteria for effective planning and instruction. Emphasizes clear content and language objectives, building background knowledge, promoting interaction, practice, application, and assessment.
Cognitive Academic Language Learning Approach (CALLA)
(Chamot & O’Malley, 1994)
Integrates content-area and language instruction with explicit attention to learning strategies.
Based on cognitive learning theory, CALLA emphasizes skills that promote active learning.

ELLs are doing two jobs at the same time: learning a new language while learning new academic content. ELLs are moving between the two worlds of their ESL classroom and their content classrooms, and they have to work harder and need more support than the average native English-speaking student who has an age-appropriate command of the English language. (Carrier, 2005, p. 6)

Lee and Fradd (Lee & Fradd, 1998, 2001; Lee, 2004) have articulated a specific framework for assisting students in the context of science. This "instructional congruence framework" emphasizes the integration of students' language and cultural experiences with content and literacy development. Such an approach should emphasize the many cultural and linguistic strengths that ELLs bring to a learning situation (Ernst-Slavit & Slavit, 2007). Johnson (2005) showed how this could be used in a multicultural seventh-grade classroom studying bioterrorism. Johnson's students drew from their political experiences, knowledge of air-borne diseases from their native countries, and stories of family members to eventually author a student handbook for responding to a bioterror event. Such a student experience is directly in line with the instructional congruence framework and employs a two-for-one teaching strategy aimed at both linguistic and content development.

Thompson and Rubenstein (2000) provided a general list of strategies for teaching mathematical vocabulary. We build on and expand that list to provide selected strategies that support ELLs as they learn mathematics, the math register, and how to "talk math."

Introduce new vocabulary in a thoughtful and integrated manner

Vocabulary is best taught not as a separate activity, but as part of the lesson. For example, students who memorize the definition of "square" without solving a problem or having discussion involving squares often make superficial meaning of this term. Manipulatives and visual aides, such as pictures, graphic organizers, charts, and bulletin boards, are good support for these conversations. It has been recommended that the introduction of new vocabulary be limited to fewer than 12 words per lesson (Fathman, Quinn, & Kessler, 1992). In addition, teachers can better communicate with their ELLs if they limit the use of idioms, speak slow, and use visuals and gestures. Breaking the lesson into smaller units and pausing and stressing key terms is also helpful.

Identify and highlight key words with multiple meanings

In addition to the problematic words and phrases discussed above, ELLs can have difficulty with words that have multiple meanings in social and academic language, or in other content areas. For example, the word "table" can refer to a "times table" for multiplication facts or a "table of values" for graphing functions. "Table" may also have very different meanings and usages in non-mathematical contexts such as "timetable" in social studies, "table of contents" in language arts, "water table" in physical science, and "periodic table" in chemistry. Identifying and carefully planning the use of any such words in a lesson can support students' efforts to follow the subsequent line of discourse.

Preview and review

This technique provides a lesson introduction (which can be given to all students or only to ELLs) via a handout, an outline of the entire lesson on the board or overhead, and a list of key words. This preview provides context for the lesson, and small-group discussion can support any of these steps. After the lesson, a review of its main aspects, including both key content and language features, can be provided to further clarify or reinforce learning goals as well as key terms. Handouts or small-group discussion could be used for this step as well.

Kristie, a middle school science teacher, makes use of this technique in all of her classes, including those with ELLs. Her use of preview is extended through the use of a "hula skirt," a piece of paper folded down the middle and cut horizontally into four or five strips on each side. The students are asked to write key terms and definitions on the left and provide a visual on the right. These terms are then used during the lesson, and the students make regular use of the hula skirt throughout. Kristie states:
For my ELLs, I always try to use different modalities to get them to understand the vocabulary. The hula skirt is kind of fun, and it gets them to write a definition and connect it to a visual. I tell them I am bad at Pictionary, you know, like stick figures and stuff, so the drawing doesn't have to be perfect. But it really connects them to the meaning of the word.

Kristie also uses the hula skirt for a Jeopardy-like game, by having one of a pair of students fold and cover the strip with the illustration, and asking the other to provide either the word or definition.

Brainstorming the meaning and origin of technical terms

Helping students brainstorm the meaning of technical words and expressions might unveil potential connections between the meaning of the word, the student's language background, and the math register. For example, discussing "degrees" as the amount it "grades out" the circular distance between the angle's rays can connect this term and idea to the Spanish word "grados." Word origins and relationships can also be helpful, such as discussing how the term hypotenuse is derived from the Greek word for "stretching under," or connecting the word "rational" to "ratio" to help make clear that all rational numbers can be expressed as a ratio of integers.

Validating students' languages and cultures

Research indicates that students' home languages can play a significant role in learning complex material, including content encountered in mathematics classrooms. This is especially true when students are afforded opportunities to incorporate their home languages into classroom discourse (Thomas & Collier, 2002). Even teachers who do not speak an ELL's home language can still make use of this strategy by affording opportunities for students to access books, handouts, or Web sites in their native language, or working with a peer or teaching assistant versed in the native language.

Arthur, a middle school teacher in a building with a large number of Mexican and Central American students, builds on students' knowledge of Spanish by using cognates—a word in one language that is similar in meaning and form to a word in another language. Arthur states, "My Spanish-speaking students understand more English than they realize. For example, they know círculo (circle), lateral (lateral; related to the side), cuadrado (a square or special quadrilateral), and even words like edificio (edifice), casi (quasi; resembling something), and creciendo (crescendo)." The use of cognates helps Arthur validate the students' first language while enabling students to learn language and content through vocabulary that can be easily identifiable in its written form.

All students come with varied lived experiences and knowledge that often leads to creative ways of solving mathematical problems. Sharing such samples of student thinking and problem solving is currently at the heart of mathematics education reform. But classrooms with ELLs can be endowed with unique perspectives on concepts or algorithms learned in another school culture or perhaps through a novel context for application of a specific mathematical topic. Such contributions could be organized through the use of writing activities or in small groups, as discussed next.

Cooperative learning and other opportunities for interaction

It is possible for students of diverse linguistic and educational backgrounds to work together on a common task in pursuit of a common goal. Collaborative groups provide opportunities for students to hear and use the math register while also developing mathematical understanding. Depending on the students' language proficiency, this works very well in groups with diverse language backgrounds, since students must use English to communicate with all the members of the group. Teachers can provide visuals with key words to support students with emerging language proficiency, even in groups with a variety of home languages.

Maddie, a seventh grade mathematics teacher with students from Mexico, Eastern Europe, and Africa, made simultaneous use of several two-for-one strategies when she asked her class to count on their fingers. Maddie noticed that Chimwala began counting with her thumb, others began with their pinky, and most with their index fingers. After this realization, Maddie asked all her students to share in groups how they use their fingers, or any other body parts, in the counting process. Though she did not choose to do so, Maddie could have extended this discussion into exploring the various algorithms for performing arithmetic on whole numbers that students bring from their various home and school cultures.

Taking risks and making mistakes

Learning a second language, including the math register, has an affective base. Students need to be encouraged to ask questions and take risks; making mistakes is a part of learning. If students' answers are not correct, or if students are not able to follow the emerging lines of discourse, patience may be needed to ensure that student risk-taking and participation will continue.


All students need support to participate in mathematical conversations, but attention to equity in a mathematics classroom must address the linguistic demands placed on ELLs; mathematical discourse is not easily accessible when presented in a second language. Learning the math register can become a complex endeavor for ELLs, because many words cannot be translated from English to their native languages, and comparable terms and parallel ways of considering ideas may not exist across languages (Lee & Fradd, 1998). This article has offered a perspective for thinking about the role of language in mathematical development, and ways in which teachers of mathematics can facilitate the two-for-one learning goal of content and linguistic development. Though not always easy to implement, the above strategies can enrich the mathematical learning experience for all students, including English language learners.