BILINGUAL MATHEMATICS TEACHING
Bilingual Mathematics Learners: How Views of Language, Bilingual Learners, and
Bilingual classs SMA Taruna Bakti and SMAK 2 BPK Penabur bandung
Jalan l.RE martadinata No. 52 Bandung and Jalan Pasirkaliki No. 157 Bandung
Understanding the relationship between language and mathematics learning is crucial to
designing mathematics instruction for students who are English Learners (ELs) and/or
bilingual.
Before we can address questions about instruction for this population, we need to
first examine views of bilingual mathematics learners and how they use language to
communicate mathematically. This chapter considers how our conceptions of bilingual
mathematics learners impact instruction for this population. In particular, I examine how views
of the relationship between mathematics and language constrain instruction. I describe three
views of bilingual mathematics learners, examine how these views impact instruction, and
critique these views using a sociocultural perspective.
Understanding bilingual mathematics learners and developing principled instruction is a
pressing practical issue, particularly for Indonesian students. An increasing number of school age
children in the Indonesia such as Bilingual class for indonesian in all big city in Indonesia.
Early studies of bilingual students learning mathematics focused on word problems,
especially translating word problems from English to mathematical symbols. Most of these
studies characterized the challenges that bilingual students faced as acquiring vocabulary or
struggling with the mathematics register. Recommendations for instruction for English learners
Bilingual Mathematics Learners
first examine views of bilingual mathematics learners and how they use language to
communicate mathematically. This chapter considers how our conceptions of bilingual
mathematics learners impact instruction for this population. In particular, I examine how views
of the relationship between mathematics and language constrain instruction. I describe three
views of bilingual mathematics learners, examine how these views impact instruction, and
critique these views using a sociocultural perspective.
Understanding bilingual mathematics learners and developing principled instruction is a
pressing practical issue, particularly for Indonesian students. An increasing number of school age
children in the Indonesia such as Bilingual class for indonesian in all big city in Indonesia.
Early studies of bilingual students learning mathematics focused on word problems,
especially translating word problems from English to mathematical symbols. Most of these
studies characterized the challenges that bilingual students faced as acquiring vocabulary or
struggling with the mathematics register. Recommendations for instruction for English learners
Bilingual Mathematics Learners
That emphasize vocabulary and reading comprehension skills reflect this focus. In contrast,
current research on mathematics learning emphasizes how students construct multiple meanings, negotiate meanings through interactions with peers and teachers, and participate in mathematical communication. Although research has explored mathematical communication as a central aspect of learning mathematics in monolingual classrooms, few studies have addressed
mathematical communication in bilingual classrooms .
The increased emphasis on mathematical communication in reform classrooms could
result in several scenarios. On the one hand, this emphasis could create additional obstacles for
bilingual learners. On the other hand, it might provide additional opportunities for bilingual
learners to flourish. And lastly, it might create a combination of these two scenarios, depending
on the classroom context. Without empirical studies that explore these hypothetical scenarios and examine mathematical communication in classrooms with bilingual students, it is impossible to reach conclusions regarding the impact of reform on bilingual learners. When carrying out these
studies or designing instruction, we need to first consider how we conceptualize language,
bilingual learners, and mathematical communication. As researchers, designers, or teachers we
can only see what our conceptual frameworks allow us to see. Our views will have great impact
on our conclusions and recommendations.
The aim of this chapter is to describe three views of bilingual mathematics learners and
explore how these views impact instruction and equity for this population. I examine three
perspectives on bilingual mathematics learners, describe how the first two constrain research and instruction, and consider how a sociocultural perspective can inform our understanding of the processes underlying learning mathematics when learning English. The first perspective
Bilingual Mathematics Learners
current research on mathematics learning emphasizes how students construct multiple meanings, negotiate meanings through interactions with peers and teachers, and participate in mathematical communication. Although research has explored mathematical communication as a central aspect of learning mathematics in monolingual classrooms, few studies have addressed
mathematical communication in bilingual classrooms .
The increased emphasis on mathematical communication in reform classrooms could
result in several scenarios. On the one hand, this emphasis could create additional obstacles for
bilingual learners. On the other hand, it might provide additional opportunities for bilingual
learners to flourish. And lastly, it might create a combination of these two scenarios, depending
on the classroom context. Without empirical studies that explore these hypothetical scenarios and examine mathematical communication in classrooms with bilingual students, it is impossible to reach conclusions regarding the impact of reform on bilingual learners. When carrying out these
studies or designing instruction, we need to first consider how we conceptualize language,
bilingual learners, and mathematical communication. As researchers, designers, or teachers we
can only see what our conceptual frameworks allow us to see. Our views will have great impact
on our conclusions and recommendations.
The aim of this chapter is to describe three views of bilingual mathematics learners and
explore how these views impact instruction and equity for this population. I examine three
perspectives on bilingual mathematics learners, describe how the first two constrain research and instruction, and consider how a sociocultural perspective can inform our understanding of the processes underlying learning mathematics when learning English. The first perspective
Bilingual Mathematics Learners
Emphasizes acquiring vocabulary, the second emphasizes multiple meanings, and the third
emphasizes participation in mathematical Discourse practices. The third perspective is a situated
and sociocultural view of language and mathematics learning that uses the concepts of registers.
I question the efficacy of the first two perspectives for understanding bilingual
mathematics learners and designing instruction for this population. The first two views can create inequities in the classroom because they emphasize what learners don’t know or can’t do. In contrast, a sociocultural perspective shifts away from deficiency models of bilingual learners and instead focuses on describing the resources bilingual students use to communicate
mathematically. Without this shift we will have a limited view of these learners and we will
design instruction that neglects the competencies they bring to mathematics classrooms. If all we
see are students who don’t speak English, mispronounce English words, or don’t know
vocabulary, instruction will focus on these deficiencies. If, instead, we learn to recognize the
mathematical ideas these students express in spite of their accents, code-switching, or missing
vocabulary, then instruction can build on students’ competencies and resources.
Below I describe three perspectives of bilingual mathematics learners: acquiring
vocabulary, constructing multiple meanings, and participating in Discourse practices.
emphasizes participation in mathematical Discourse practices. The third perspective is a situated
and sociocultural view of language and mathematics learning that uses the concepts of registers.
I question the efficacy of the first two perspectives for understanding bilingual
mathematics learners and designing instruction for this population. The first two views can create inequities in the classroom because they emphasize what learners don’t know or can’t do. In contrast, a sociocultural perspective shifts away from deficiency models of bilingual learners and instead focuses on describing the resources bilingual students use to communicate
mathematically. Without this shift we will have a limited view of these learners and we will
design instruction that neglects the competencies they bring to mathematics classrooms. If all we
see are students who don’t speak English, mispronounce English words, or don’t know
vocabulary, instruction will focus on these deficiencies. If, instead, we learn to recognize the
mathematical ideas these students express in spite of their accents, code-switching, or missing
vocabulary, then instruction can build on students’ competencies and resources.
Below I describe three perspectives of bilingual mathematics learners: acquiring
vocabulary, constructing multiple meanings, and participating in Discourse practices.
I argue that the third view, a sociocultural perspective, enriches our views of the relationship between language and learning mathematics, expands what counts as competence in mathematical communication, and provides a basis for designing equitable instruction. To make this case, I first compare and contrast the three perspectives and then present two examples to substantiate my claims regarding the contributions of a sociocultural perspective.
Acquiring Vocabulary
One view of bilingual mathematics learners is that their main challenge is acquiring
vocabulary. This first perspective defines learning mathematics as learning to carry out
computations or solve traditional word problems, and emphasizes vocabulary as the central issue
for English learners as they learn mathematics. This view is reflected in early research on
bilingual mathematics learners that focused primarily on how students understood individual
vocabulary terms or translated traditional word problems from English to mathematical symbols
. Recommendations for mathematics instruction for English learners have also emphasized vocabulary and reading comprehension Although an emphasis on vocabulary and reading comprehension may have beensufficient in the past, this emphasis does not match current views of mathematical proficiency or the activities in contemporary classrooms. In many mathematics classrooms today, the main activities are not carrying out arithmetic computations, solving traditional word problems, reading textbooks, or completing worksheets. Many students participate in a variety of oral and written practices such as explaining solution processes, describing conjectures, proving conclusions, and presenting arguments. As a consequence, reading and understanding mathematical texts or traditional word problems are no longer the best examples of how language and learning mathematics intersect.
Even in traditional classrooms where there may be little oral discussion, learning
mathematical language involves more than learning vocabulary: words have multiple meanings,
vocabulary. This first perspective defines learning mathematics as learning to carry out
computations or solve traditional word problems, and emphasizes vocabulary as the central issue
for English learners as they learn mathematics. This view is reflected in early research on
bilingual mathematics learners that focused primarily on how students understood individual
vocabulary terms or translated traditional word problems from English to mathematical symbols
. Recommendations for mathematics instruction for English learners have also emphasized vocabulary and reading comprehension Although an emphasis on vocabulary and reading comprehension may have beensufficient in the past, this emphasis does not match current views of mathematical proficiency or the activities in contemporary classrooms. In many mathematics classrooms today, the main activities are not carrying out arithmetic computations, solving traditional word problems, reading textbooks, or completing worksheets. Many students participate in a variety of oral and written practices such as explaining solution processes, describing conjectures, proving conclusions, and presenting arguments. As a consequence, reading and understanding mathematical texts or traditional word problems are no longer the best examples of how language and learning mathematics intersect.
Even in traditional classrooms where there may be little oral discussion, learning
mathematical language involves more than learning vocabulary: words have multiple meanings,
Meanings depend on situations, and learning to use mathematical language requires learning
whn to use different meanings. Vocabulary (along with decoding) is certainly an aspect of
developing reading comprehension at the word level. However, vocabulary is not sufficient for
becoming a competent reader. Reading comprehension involves skills beyond the word level,
such as constructing meaning from text, using metacognitive strategies, and participating in
academic language practices .
An emphasis on vocabulary provides a narrow view of mathematical communication.
This narrow view can have a negative impact on assessment and instruction for bilingual
learners. English oral proficiency can affect how teachers assess a student’s mathematical
competence. For example, if we focus only on a student's failure to use the correct word, we can
miss the student’s competency in making conjectures, constructing arguments, addressing special cases, or dealing with contradictory evidence. If we conceive of “language” as only vocabulary, we are limiting the scope of communicative activities used to assess mathematical competence, and many students will appear less competent. Instruction focusing on low-level linguistic skills, such as vocabulary, neglects the more complex language skills necessary for learning and doing mathematics.
Lastly, this view perpetuates a deficiency model of bilingual learners that can have a
negative impact on English learners’ access to mathematical instruction. English learners may
have a smaller or less accurate mathematical vocabulary in English than native English speakers.
We can see this as a deficiency or we can notice this difference while also noticing other
competencies for communicating mathematically. “Vocabulary” need not be construed as a
deficiency, a reason for remedial instruction, or a pre-requisite that bilingual learners must
achieve before they can participate in more conceptual or advanced mathematics instruction.
English learners can learn vocabulary at the same time they participate in many types of lessons,
including conceptual mathematical activities.
Constructing Multiple Meanings
A second perspective on bilingual mathematics learners describes learning mathematics
as constructing multiple meanings for words. Work in mathematics education from this
perspective has used the notion of the mathematics register.
A second perspective on bilingual mathematics learners describes learning mathematics
as constructing multiple meanings for words. Work in mathematics education from this
perspective has used the notion of the mathematics register.
Multiple meanings can create obstacles in mathematical conversations because students
often use colloquial meanings while the teacher (or other students) may use mathematical
meanings. For example, the word “prime” can have different meanings depending on whether it
is used to refer to “prime number,” “prime time,” Another example the term segitiga sama kaki refers to an equlateral triangles not a same leg triangles because in indonesian languange 'kaki' means 'leg' in english.
often use colloquial meanings while the teacher (or other students) may use mathematical
meanings. For example, the word “prime” can have different meanings depending on whether it
is used to refer to “prime number,” “prime time,” Another example the term segitiga sama kaki refers to an equlateral triangles not a same leg triangles because in indonesian languange 'kaki' means 'leg' in english.
Mathematical Discussions
In this section I examine two mathematical discussions to illustrate the limitations of the
vocabulary and multiple meanings perspectives and to describe how a sociocultural perspective
enriches our view of language, provides an alternative to deficiency models of learners, and
generates different questions for both reseacrh and instruction. I selected the first example to
illustrate the limitations of the vocabulary perspective and the second example to illustrate the
limitations of the multiple meanings perspective. The two examples presented below show the
complexity that using a situated and sociocultural perspective as an analytical lens brings to the
study of bilingual mathematics learners. The first example shows us how the vocabulary
perspective fails to capture students’ competencies in communicating mathematically. The
second example shows that the multiple meanings perspective can also fall short of a full
description of the resources that students use.
In presenting these examples, I also show how to use a sociocultural perspective to
identify student competencies and resources that instruction can build on to support mathematics learning.
vocabulary and multiple meanings perspectives and to describe how a sociocultural perspective
enriches our view of language, provides an alternative to deficiency models of learners, and
generates different questions for both reseacrh and instruction. I selected the first example to
illustrate the limitations of the vocabulary perspective and the second example to illustrate the
limitations of the multiple meanings perspective. The two examples presented below show the
complexity that using a situated and sociocultural perspective as an analytical lens brings to the
study of bilingual mathematics learners. The first example shows us how the vocabulary
perspective fails to capture students’ competencies in communicating mathematically. The
second example shows that the multiple meanings perspective can also fall short of a full
description of the resources that students use.
In presenting these examples, I also show how to use a sociocultural perspective to
identify student competencies and resources that instruction can build on to support mathematics learning.
Example 1: Describing a Pattern
A group of seventh and eighth grade students in a summer mathematics course
constructed rectangles with the same area but different perimeters and looked for a pattern to
relate the dimensions and the perimeter of their rectangles. Below is a problem similar to the one
they were working on:
1. Look for all the rectangles with area 36 and write down the dimensions.
2. Calculate the perimeter for each rectangle.
3. Describe a pattern relating the perimeter and the dimensions.
In this classroom, there was one bilingual teacher and one monolingual teacher. A
group of four students were videotaped as they talked in their small group and with the
bilingual teacher . They attempted to describe the pattern in their group
and searched for the bahasa indonesia word for rectangle. The students produced several suggestions,
including persegipanjang , segitiga, jajaran genjang, and persegi. Although these students
attempted to find a term to refer to the rectangles neither the teacher nor the other students
provided the correct Bahasa Indonesia word, Persegi panjang [rectangle].
Later on, a second teacher (monolingual English speaker) asked several questions from
the front of the class. In response, one of the students in this small group, described a
objects). This move also shifts our attention from words to mathematical ideas, as expressed not
only through words but also other modes. This shift is particularly important to uncover the
mathematical competencies for students who are learning English.
Example 2: Clarifying a Description
While the first example fits the expectation that bilingual students struggle with
vocabulary, the vocabulary perspective was not sufficient to describe that student’s competence.
The second example highlights the limitations of the vocabulary perspective for describing
mathematical communication and shows how code switching can be a resource for bilingual
speakers. In the following discussion two students used both languages not for vocabulary, but to
clarify the mathematical meaning of a description.
Bilingual learners may be different than monolinguals but they should not be
defined by deficiencies.
The two examples illustrate several aspects of learning mathematics in a bilingual
classroom that only become visible when using a sociocultural perspective:
1) Learning to participate in mathematical Discourse is not merely or primarily a matter
of learning vocabulary. During conversations in mathematics classrooms students are
also learning to participate in valued mathematical Discourse practices such as
describing patterns, making generalizations, and using representations to support
claims.
2) Bilingual learners use many resources to communicate mathematically: gestures,
objects, everyday experiences, their first language, code switching, and mathematical
representations.
3) There are multiple uses of Bahasa Indonesia in mathematical conversations between bilingual
students. While some students use Bahasa Indonesia to label objects, other students use Bahasa
to explain a concept, justify an answer, or elaborate on an explanation or description.
4) Bilingual students bring multiple competencies to the classroom. For example, even a
student who is missing vocabulary may be proficient in describing patterns, using
mathematical constructions, or presenting mathematically sound arguments.
A sociocultural perspective points to several aspects of classroom instruction that need
to be considered. Classroom instruction should support bilingual students' engagement in
conversations about mathematics, going beyond translating vocabulary and involving students
in communicating about mathematical ideas.
While the first example fits the expectation that bilingual students struggle with
vocabulary, the vocabulary perspective was not sufficient to describe that student’s competence.
The second example highlights the limitations of the vocabulary perspective for describing
mathematical communication and shows how code switching can be a resource for bilingual
speakers. In the following discussion two students used both languages not for vocabulary, but to
clarify the mathematical meaning of a description.
Bilingual learners may be different than monolinguals but they should not be
defined by deficiencies.
The two examples illustrate several aspects of learning mathematics in a bilingual
classroom that only become visible when using a sociocultural perspective:
1) Learning to participate in mathematical Discourse is not merely or primarily a matter
of learning vocabulary. During conversations in mathematics classrooms students are
also learning to participate in valued mathematical Discourse practices such as
describing patterns, making generalizations, and using representations to support
claims.
2) Bilingual learners use many resources to communicate mathematically: gestures,
objects, everyday experiences, their first language, code switching, and mathematical
representations.
3) There are multiple uses of Bahasa Indonesia in mathematical conversations between bilingual
students. While some students use Bahasa Indonesia to label objects, other students use Bahasa
to explain a concept, justify an answer, or elaborate on an explanation or description.
4) Bilingual students bring multiple competencies to the classroom. For example, even a
student who is missing vocabulary may be proficient in describing patterns, using
mathematical constructions, or presenting mathematically sound arguments.
A sociocultural perspective points to several aspects of classroom instruction that need
to be considered. Classroom instruction should support bilingual students' engagement in
conversations about mathematics, going beyond translating vocabulary and involving students
in communicating about mathematical ideas.
It is not a question of whether or not students should learn vocabulary but rather how
instruction can best support students learning both vocabulary and mathematics. Vocabulary
drill and practice is not the most effective instructional practice for learning either vocabulary
or mathematics. Instead, vocabulary and second language acquisition experts describe
vocabulary acquisition in a first or second language as occurring most successfully in
instructional contexts that are language rich, actively involve students in using language,
require both receptive and expressive understanding, and require students to use words in
multiple ways over extended periods of time.
Understanding the mathematical ideas in what students say and do can be difficult
when teaching, perhaps especially so when working with students who are learning English. It
may not be easy (or even possible) to sort out which aspects of a student's utterance are results
of the student's conceptual understanding or the student's English proficiency. However, if the
goal of instruction is to support students as they learn mathematics, determining the origin of
an error is not as important as listening for students’ mathematical ideas and uncovering the
mathematical competence in what they are saying and doing. Hearing mathematical ideas and
uncovering mathematical competence is only possible if we move beyond limited views of
language and deficiency models of bilingual learners.
instruction can best support students learning both vocabulary and mathematics. Vocabulary
drill and practice is not the most effective instructional practice for learning either vocabulary
or mathematics. Instead, vocabulary and second language acquisition experts describe
vocabulary acquisition in a first or second language as occurring most successfully in
instructional contexts that are language rich, actively involve students in using language,
require both receptive and expressive understanding, and require students to use words in
multiple ways over extended periods of time.
Understanding the mathematical ideas in what students say and do can be difficult
when teaching, perhaps especially so when working with students who are learning English. It
may not be easy (or even possible) to sort out which aspects of a student's utterance are results
of the student's conceptual understanding or the student's English proficiency. However, if the
goal of instruction is to support students as they learn mathematics, determining the origin of
an error is not as important as listening for students’ mathematical ideas and uncovering the
mathematical competence in what they are saying and doing. Hearing mathematical ideas and
uncovering mathematical competence is only possible if we move beyond limited views of
language and deficiency models of bilingual learners.
Conclution :
1. Language: Mathematicalcommunication involvesmore than words, registers,
or multiple meanings; it alsoinvolves non-languageresources and discoursepractices.
2. Bilingual learners: Whilebilingual learners are
different than monolinguals,they are not deficient; theybring competencies and use
resources. Thesecompetencies and resourcesmay be the same or different
than monolinguals.
3. Instruction should focus onuncovering studentcompetencies and resources
and building on these.
or multiple meanings; it alsoinvolves non-languageresources and discoursepractices.
2. Bilingual learners: Whilebilingual learners are
different than monolinguals,they are not deficient; theybring competencies and use
resources. Thesecompetencies and resourcesmay be the same or different
than monolinguals.
3. Instruction should focus onuncovering studentcompetencies and resources
and building on these.
the end.
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