Equity
language
for
learners
By Rusty Bresser, Kathy Melanese, and Christine Sphar
Stefanie Timmermann/iStockphoto.com (2)
Learn how to focus
mathematical language
on concepts to
accommodate the needs
of the 10 percent of U.S.
students whose first
language is not English.
Copyright © 2009 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
E
veryone uses language to learn mathematics. Paying close attention to the needs
of students who are learning English as
a second (or third) language is crucial so that
we can modify lessons to accommodate those
needs. The Equity Principle requires that we
accommodate differences in our diverse student
population to help everyone learn mathematics
(NCTM 2000).
About 10 percent of the nation’s public
school students are English Language Learners
(ELLs), and every year this percentage increases
(NCELA 2007). Although language arts is probably the most challenging subject of their school
day, mathematics is likely a struggle for these
students as well. In fact, achievement data show
that ELLs are not performing at the same levels
in mathematics as their native English-speaking
counterparts (NAEP 2007).
Teaching ELLs
Many educators share the misconception that
because mathematics uses symbols, it is not
associated with any language or culture and
is ideal for facilitating the transition of recent
immigrant students into English instruction
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(Garrison 1997). On the contrary, language plays
an important role in learning mathematics.
We use language to explain mathematical concepts and carry out mathematical procedures.
Students can deepen and cement their understanding of mathematics by using language to
communicate and reflect on their ideas. When
students converse about their mathematical
thinking, such talk can help them improve their
ability to reason logically (Chapin and Johnson
2000).
Communication in mathematics has the
potential to facilitate understanding, but the
practice of discussing ideas in English may place
children who are learning English as a second
language at a distinct disadvantage, especially
if they do not fully understand key vocabulary
words, or if they have an incomplete understanding of syntax and grammar.
The challenge of teaching mathematics to
English language learners lies not only in making lessons comprehensible to students but also
in ensuring that students have the language
needed to understand instruction and express
their grasp of mathematical concepts. ELLs have
the dual task of learning the second language
teaching children mathematics • October 2009 171
and the subject content simultaneously. For this
reason, it is important to set both language and
content objectives for ELLs. Just as language
may not develop if we focus only on subject
matter, content knowledge cannot grow if we
focus only on learning the English language (Hill
and Flynn 2006).
Many teachers use supports such as manipulatives, visuals, and graphics to help students
understand the content in their math lessons.
These supports are important for building basic
understanding of mathematical concepts but
may not provide students with enough linguistic support for them to discuss their thinking,
which would lead to deeper understanding of
content. For example, if a student’s understanding of polygons is based on a two-column chart
with drawings that distinguish polygons from
shapes that are not polygons, once the chart is
put away, the student may not have internalized enough of the lesson’s linguistic
elements to be able to continue learning
subsequent lessons on polygons. Having
the language to talk about mathematical concepts is crucial to developing
an understanding of those concepts
(Bresser, Melanese, and Sphar 2008).
The challenge for teachers is to focus
on mathematical concepts and the
academic language that is specific to
mathematics. Teachers must be aware
of the linguistic demands of their lessons
and how they will address those demands
explicitly during instruction so that ELLs
can fully participate. During the lesson, will
students be expected to do the following:
• Compare and contrast numbers, data sets, or
geometric figures.
• Hypothesize while playing a number guessing
game.
• Describe the location of ordered pairs on a
Cartesian coordinate plane.
• Summarize a strategy they used to solve a
multiplication problem.
Each of these activities requires students to
use language to communicate their mathematical thinking. And if students are expected to use
these language functions in math class while
they are learning English as a second language,
they will need well-designed extra support.
172
October 2009 • teaching children mathematics Modifying math lessons
During the past four years, we have collaborated on designing lessons to help all students,
especially ELLs, accomplish two goals: develop
English proficiency and develop mathematical
understanding. Using existing mathematics
lessons intended for native English speakers,
we combined our knowledge and experience
of English language development and of mathematics to modify lessons in ways that help us
explicitly structure experiences to benefit all
students, especially ELLs. When we modified
lessons, we used the following questions to
guide our work:
1. What is the lesson’s mathematical goal, and
what language goal will support students’
understanding of this goal?
2. How can we support students when they are
required to talk about their mathematical
thinking?
3. How can we accommodate students with different levels of English language proficiency?
4. Are there vocabulary words with which students are unfamiliar, and how can we explicitly teach these words?
5. In what ways can we build in opportunities for
discussion?
Determining the language goal
We based the language goals that we identified on the content of the mathematics lessons. After determining what we would expect
students to do in each lesson, we analyzed the
lesson for the language they would need to fully
participate. This included vocabulary and language that teachers would use during instruction, as well as language that students would
need to use to let us know if they had met our
mathematical goals.
For example, if students are working toward
the goal of determining equivalent fractions,
they need language to describe the fractions
they create as well as language to compare and
contrast them with other fractions.
If the mathematics goal is that students solve
a word problem using an effective strategy that
makes sense to them, then they must be able
to express, orally or in writing, the sequence of
steps that they used to solve the problem.
If students are learning about classifying
and categorizing geometric shapes, the lanwww.nctm.org
table 1
We used sentence frames for dice game predictions.
Function
Beginning
Predicting
The
have
Examples
I will roll a six.
will
.
Intermediate
Advanced
I predict that
will
.
I predict that
will
because
.
I predict that I
will roll a six.
I predict that I will roll a six because
I have rolled it more times than any other
number.
guage goal would be for students to describe
the categories they chose for the shapes and to
explain their reasoning.
Each of these three mathematical goals—
learning about fractions, solving word problems, and classifying and categorizing geometric shapes—has an accompanying logical
language goal (or goals). In these instances, the
respective language goals were to describe, compare and contrast, sequence, and categorize.
With prompts such as the following from the
teacher, native English-speaking children can
perform these language functions to articulate
their mathematical understanding: “Tell me
what knowledge you have about fractions.”
“Explain the steps you took to solve this
word problem.”
“Tell me what category this shape belongs
to, and why.”
ELLs may understand lesson content, but
inexperience with the language can keep them
from articulating what they know. Their struggles with the language of instruction can also
lead them to partial or inaccurate understanding of the content. Until they verbalize their
understanding, what they have or have not
learned remains a mystery to the teacher and
may even be unclear to students themselves.
Choosing a language goal or language function that matches the mathematical content
goal makes the learning more observable to all
(Bresser et al. 2008).
Once we determined the mathematical
and language goals for a particular lesson, we
designed sample ways that students could
frame sentences to practice expressing their
thinking about a mathematical concept with
a partner, during small-group time, and during whole-class discussions. For example, after
explicitly teaching key vocabulary terms such
as predict, likely, unlikely, equally likely, and
chance, we introduced sentence frames to help
students make predictions (see Table 1) on the
basis of data that they collected during a dice
game.
If students are sequencing the steps that
they would use to solve a subtraction problem,
they might use this sentence frame:
Supporting classroom math talk
Such frames can be powerful tools in the
hands of students learning the English language. When structured appropriately, the
frames are flexible enough to be used in a
variety of contexts. They allow students to use
key vocabulary terms to put together complete
thoughts that can be connected, confirmed,
rejected, revised, and understood.
Just because we provide opportunities for
ELLs to talk about their mathematical ideas in
English does not necessarily mean that they
will have the language to do so. Students must
understand a lesson’s academic vocabulary,
and they must also be able to use that vocabulary in their conversations during the lesson.
www.nctm.org
First I
, then I
.
First I said, Eight, then I counted back three
to get five.
If students are comparing and contrasting
quadrilaterals, they might use this frame:
A
has
and
, but a
has
both have
.
.
A square has two pairs of parallel sides, but
a trapezoid has one pair of parallel sides.
A square and a trapezoid both have four
sides.
teaching children mathematics • October 2009 173
Accommodating various proficiencies
The classrooms in which we worked included
native English speakers and learners at different
levels of English language acquisition: beginning,
intermediate, and advanced. When modifying
math lessons, we differentiated instruction by taking these levels into account and creating frames
that were increasingly more sophisticated.
While helping fourth graders compare circle
circumferences and diameters, we decided that
beginning-level students could talk about circumference and diameter using this frame:
The
is
. The
is
Or students might say the following:
The diameter is three centimeters.
The circumference is nine centimeters.”
We wanted intermediate-level students to
use comparative language, so we modified the
frames for them:
The
is
than
.
The circumference is longer than the
diameter.
.
Or they could use this frame:
The diameter is short.
The circumference is long.
Reflect and discuss:
“Equity for language learners”
Reflect on the following questions related to “Equity for language learners” by Rusty
Bresser, Kathy Melanese, and Christine Sphar. The prompts are provided as a tool to
aid you in reflecting independently on the article, discussing it with your colleagues,
and considering how the authors’ ideas might benefir your own clasroom practice.
1. What role does language play in learning mathematics?
2. List some challenges that English Language Learners (ELLs) face during math
instruction.
3. State important points to remember when modifying a math lesson for ELLs.
4. How can teachers differentiate math instruction for ELLs with varying levels of
proficiency in English?
5. What are the benefits of using sentence frames to teach mathematics?
With the following questions in mind, consider a math lesson you plan to teach:
1. What is the lesson’s mathematical goal?
2. Which vocabulary terms must students understand and use? How can key
vocabulary be explicitly taught?
3. What will students be able to say if they meet the math goal? In other words,
state the terminology that students will use during the lesson.
4. For what purpose will students use language (e.g., to describe, to categorize, to
hypothesize, to sequence, to compare and contrast)?
5. What strategies will you use to help ELLs understand mathematical content and
generate language?
6. How will you differentiate the lesson for students whose English language proficiency levels vary?
7. Are opportunities for discussion built into the lesson?
Tell us how you used “Reflect and discuss” as part of your professional development.
Submit letters to Teaching Children Mathematics at [email protected]. Include “readers exchange” in the subject line. Find more information at tcm.msubmit.net.
174
October 2009 • teaching children mathematics The
is
times
than
.
The circumference is three times longer
than the diameter.
Advanced students and native English speakers need frames that help them make more complex statements:
The
of the
is about/around/approximately
.
In addition to creating frames, we differentiated our questions to allow ELLs at all levels of
English proficiency to respond and communicate their mathematical thinking. Students
at beginning levels need simple prompts and
questions that do not require lengthy explanations. We gave them opportunities to use
physical rather than verbal responses:
• Show me the polygon.
• Touch the triangle.
Questions with a yes or no answer are also
appropriate for beginning ELLs: “Is forty-five
greater than sixty-seven?”
“Is the square a polygon?”
To provide support, teachers can build in
answers when asking short-answer questions:
• Which shape belongs to this group, the triangle or the square?
• Is this the diameter or the circumference?
• Do more people on our graph like vanilla ice
cream or chocolate ice cream?
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table 2
Before students can use a sentence frame, they must learn key vocabulary terms
that they can place in the frame to begin to talk about their learning.
Function
Describing
Beginning
A
is/has
.
Examples
A rhombus has
four angles.
Intermediate
A
is/has
, and
Advanced
,
A
might have
or
but it will always have
.
.
A square has four
sides, four vertices,
and no curves.
Students with intermediate and advanced
proficiency levels need less support to understand and respond to questions from the
teacher, but carefully designed questions can
elicit responses that reveal students’ mathematical thinking and allow for further development
of academic English. For example, instead of
asking an intermediate-level student, “What can
you tell me about the graph?” you might phrase
your question this way: “What do the most (or
the fewest) people on the graph prefer?”
This second question structure models an
answer using academic discourse: “Most people
prefer to swim. The fewest number of people
prefer to ride bikes.” Compare this answer to
the likely response to the first question: “A lot of
people like to swim.”
Students with advanced fluency can respond
to even more open-ended questions and prompts
such as, “Describe to me the steps you used to solve
the problem and explain how you used them.”
Teaching key vocabulary
,
A polygon might have three sides or six sides,
but it will always have straight sides.
Using sentence frames in primary grades
In the primary grades, literate children are
able to read different sentence frames and
choose among them independently. Students
with emergent literacy levels also make use of
the frames, but in an auditory manner. As the
teacher uses a frame, students hear how to
structure their verbal responses. The teacher
writes the frame or frames on the board to
make sure she incorporates and models the
vocabulary and syntax that will help students
articulate their thinking. When the teacher
introduces and models the frame aloud, it
helps language learners to focus on the key
vocabulary and learn how to structure some
words around the key vocabulary in order to say
something about the concept. This very careful
and intentional use of speech by the teacher is
what promotes oral language development, an
important building block for literacy.
For a lesson in which second graders had to
sort and categorize objects by their attributes,
Before students are asked to use a sentence
frame, they learn key vocabulary terms that they
can place in the frame to begin to talk about
their learning. For instance, during a third-grade
lesson on polygons, we explicitly taught key
mathematical terms such as polygon, sides, vertex, vertices, open, closed, square, and triangle in
the context of familiar objects in the classroom
(see table 2). Then we introduced the sentence
frames and had students practice describing
familiar objects: “A calendar has four sides.”
“A piece of paper has four sides, four vertices,
and is a closed shape.”
After practicing the vocabulary words and
sentence frames in familiar contexts, students
were ready to use academic language to learn
about polygons.
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teaching children mathematics • October 2009 175
• Meet the mathematical goal.
• Meet the language goal.
• Participate more than you normally do in
math lessons.
• Respond as if our sentence frames were
appropriate for the lesson (not as if we used
too many or too few frames).
we first introduced names of objects (e.g.,
blocks, metal washers, rubber bands, crayons)
that we wanted students to sort. Then we brainstormed with the students to identify some
attributes they might use to categorize the
objects (see table 3).
Whether designing sentence frames for primary- or intermediate-level students, we had
to keep in mind our chosen language goal or
function that supported the mathematical goal.
We molded the frames to fit the grade level and
the particular lesson, and we considered the
number of blanks necessary for each frame. We
also had to think about what tense we wanted
students to articulate their thinking in: past,
present, future, or conditional. After creating
the frames, we practiced using them to see if
they made sense. The most important idea that
guided us when creating sentence frames was
to keep them as open and flexible as possible to
allow students to express their own mathematical thinking. Modeling the sentence frames for
students and giving them time to practice the
frames were keys to students’ success.
Reflecting on lesson
modifications
ta ble 3
After modifying and then teaching each math
lesson, we took time to reflect on how successful the lesson was and what changes could
make it more successful. We wanted ELLs in the
class to do the following:
176
Guided by the idea that true reflection leads
to action (Freire 1970), we used the questions as
a way to assess the effectiveness of our teaching so that we could make specific changes
that would benefit all students. Our goal was
to level the playing field so that everyone had
equal access to the mathematical content
being taught.
Modifying lessons—to make them comprehensible and to provide language support
to help English learners think about new concepts, experiment with their knowledge, and
solidify their understanding—is not easy. This
is the type of work, however, that promotes
equity and helps all students, especially English
language learners, to fully participate in their
learning community and fully benefit from our
teaching.
References
Bresser, Rusty, Kathy Melanese, and Christine
Sphar. Supporting English Language Learners
in Math Class, Grades K–2. Sausalito, CA: Math
Solutions Publications, 2008.
———. Supporting English Language Learners in
Math Class, Grades 3–5. Sausalito, CA: Math
Solutions Publications, 2008.
Chapin, Suzanne H., and Art Johnson. Math Matters: Understanding the Math You Teach. Sausalito, CA: Math Solutions Publications, 2000.
Dutro, Susana, and Carrol Moran. “Rethinking
Sentence frames can be used to identify some attributes and then sort objects.
Function
Beginning
Intermediate/Advanced
Categorizing
Expect beginning ELLs to use singleword responses to describe how
objects are sorted.
These are
; these are
These objects are
These objects are
.
Examples
Blue! Red!
White!
Not white!
These are blue; these are red.
These objects are straight.
These objects are not straight.
October 2009 • teaching children mathematics .
www.nctm.org
English Language Instruction: An Architectural
Approach.” In English Learners: Reaching the
Highest Levels of English Literacy, edited by Gilbert G. García. Newark: International Reading
Association, 2003.
Farrell, Thomas S. C. Reflective Practices in Action:
80 Reflection Breaks for Busy Teachers. Thousand Oaks, CA: Corwin Press, 2000.
Freire, Paulo. Pedagogy of the Oppressed. New
York: The Seabury Press, 1970.
Garrison, Leslie. “Making the NCTM’s Standards
Work for Emergent English Speakers.” Teaching Children Mathematics 4 (November 1997):
132–38.
Hill, Jane D., and Kathleen M. Flynn. Classroom
Instruction That Works with English Language
Learners. Alexandria, VA: Association of Supervision and Curriculum Development, 2006.
National Assessment of Educational Progress
(NAEP). 2007. http://nationsreportcard.gov/
math_2007/m0015.asp.
National Council of Teachers of Mathematics
www.nctm.org
(NCTM). Principles and Standards for School
Mathematics. Reston, VA: NCTM, 2000.
National Clearinghouse for English Language
Acquisition (NCELA). 2007. www.ncela.gwu.
edu/policy/states/reports/statedata/2005LEP/
GrowingLEP_0506.pdf.
Rusty Bresser,
[email protected],
lectures and supervises
teacher education in
the Education Studies
Program of the University of California at San Diego. Kathy Melanese,
[email protected], is a Distinguished Bilingual Teacher in Residence
in the Education Studies Program
at the same university. Christine Sphar, ccsmith@
sdcoe.net, is a beginning teacher support administrator and consulting teacher in San Diego’s El Cajon
Valley School District.
teaching children mathematics • October 2009 177