Mathematical vocabulary
The math register includes a variety of words, phrases, and expressions, which
can be placed into four categories (Figure 1). Highfrequency
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 ErnstSlavit
and Slavit, 2007)
Type

Description

Examples

Highfrequency
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. WongFillmore & 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.

Representing
information in nonlinguistic 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=5b, when it
should be a=b5.
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.
Title

Approach

Description

Contentbased 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.

SpeciallyDesigned Academic
Instruction in English (SDAIE) (DiazRico & Weed, 2006) 
Teaching of gradelevel
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)
www.projectglad.com/ 
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 selfevaluation 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 contentarea 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 Englishspeaking student who has an ageappropriate
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 (ErnstSlavit & Slavit,
2007). Johnson (2005) showed how this could be used in a multicultural
seventhgrade classroom studying bioterrorism. Johnson's students drew from
their political experiences, knowledge of airborne 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
twoforone 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 nonmathematical 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 smallgroup 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 smallgroup 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 Jeopardylike 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 Spanishspeaking 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 twoforone 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 risktaking and participation will continue.
Conclusions
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
twoforone 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.
I have read this blog, very nice information but one thing is missing here that is define rational numbers which is the most important term of mathematics, so please share it.
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