Mathematics and ESL - Center for Teaching and Learning

Mathematics and ESL: Common Ground and
Uncommon Solutions
Prabha Betne (Mathematics) and
Carolyn Henner Stanchina (Education and Language Acquisition)
This article emerges from our collaboration on
a series of workshops for middle and high
school teachers of mathematics struggling to
teach English language learners (ELLs) who are,
in turn, struggling to pass the Mathematics A
Regents examination. It represents the initiation
of a conversation about the crucial role of
language comprehension in mathematics, the
parallels between second language and mathematics
acquisition and pedagogy, the roles of
teachers, and the immense challenges facing
an education system serving over 180,000 students
classified as ELLs by the National Clearinghouse
for English Language Acquisition, and
speaking some 165 languages as reported by
the New York City Department of Education.
Standards
During the 1990s, academic standards for high
school graduation were revised, requiring all
high school students, including English language
learners, to pass five Regents examinations,
including Mathematics, with a score of
65 or above. Simultaneously, the Mathematics
Regents has undergone a major shift in focus
from de-contextualized manipulation skills to
contextualized problem-solving strategies. The
common underlying assumption that math
does not depend on language, that students can
excel in math without language, no longer
holds true. The new standard of mathematical
literacy pre-supposes advanced language proficiency,
and while this shift in focus is a positive
reform, it is clear that the more mathematics
is embedded in language, the more it challenges
English language learners.
Our Questionnaire Results
For the most part, these changes in assessment
have not resulted in the type of collaborative
dialogue between ESL and Mathematics teachers
that could empower and enable Mathematics
teachers to cope with the special learning
needs of English Language Learners. Analysis
of a questionnaire we distributed to participants
before the workshops indicated that
while some teachers had developed strategies
for addressing the students’ linguistic and cultural
barriers to learning, many seemed to lay
blame on the students for lacking critical thinking
skills, lacking practice in math, lacking the
ability to identify what was being asked in a given
problem or to understand teacher explanations,
and for demonstrating an “unwillingness
to catch up in English reading abilities.”
What Makes the Language of Math So
Difficult?
It is clear that teachers need to be sensitized to
the challenge ELL students face when reading
word problems and learning mathematics. In
an attempt to contextualize math, writers have
used concepts that may be culture-specific;
therefore, they are not part of the ELL students’
schema knowledge, and not transparent enough
to allow students to guess in context. The resulting
word problems may deal with the Kentucky
Derby, financial investments, high school
proms, baseball and other contexts which may
be unfamiliar to students, thereby impairing
their reading comprehension. Because the language
of math lacks the redundancy and paraphrase
of natural language (Dale and Cuevas
338), there are few clues to meaning. Again,
normally activated reading strategies such as
guessing in context, inferring and predicting,
may not be transferable to reading in math.
In addition, lexical, syntactic and semantic
challenges abound in word problems, according
to an analysis by Dale and Cuevas (332).
Besides the specialized vocabulary, there are
long noun phrases, complex collocations, confusing
prepositions (“by” can be used to signal
multiplication, addition and division), and cul-
Mathematics and ESL • 1

tural differences in the symbolic denotations
of mathematical processes. Also, while certain
words designate specific math operations, (“less
than” may indicate subtraction), the opposite
may also be true: Hilda has 7 records. She has
5 records less than David. How many records
does David have? Since David has 7+5, “less
than” indicates addition. (Chamot and O’Malley
229).
Syntactically, students must understand
that they cannot read mathematical sentences
as if they were natural language sentences. In
algebra, for example, Crandall, Dale, Rhodes
and Spanos found “recurring errors in translations
of the language of word problems into
solution equations,” (Dale and Cuevas 334),
indicating that students map the surface syntax
of the problem statements onto their equations.
Students translated the sentence, “The
number a is five less than the number b” as
a = 5 - b, instead of the correct translation
a = b - 5. (Dale and Cuevas 335). Similarly,
given the problem, “There are 6 times as many
students as professors in the math department.
Write an equation that represents this statement,”
students typically write 6S=P because
they follow the literal word order of the natural
language sentence (Cocking and Mestre
216).
The frequent use of complex structures
which are traditionally taught at higher levels
of ESL, such as comparatives: “That car costs 5
times as much as mine does,” passive voice: “X
is defined to be equal to 0” (Dale and Cuevas
334), and logical connectors which set up various
relationships between parts of a text –
including cause-effect, similarity, and logical
sequence: “if…then, if and only if, such that...”
– also contributes to the level of linguistic difficulty.
Semantically, to understand mathematical
language, students must be familiar with discourse
devices which signal reference in English.
In the example provided by Dale and
Cuevas – “The sum of two numbers is 77. If the
first number is 10 times the other, find the number.”
– students need to understand that there
are 2 numbers, that “the first number” refers
to one of those numbers, and the elliptical “the
other” refers to the second number, whose value
they have to find (336).
What makes the language of math so difficult,
then, is both the students’ lack of a cultural
framework on which to build an understanding
of the context of a problem, and their lack
of familiarity with the discourse features of
math texts. This brief description of the challenges
inherent in solving mathematical word
problems reflects the interactive nature of the
learning process itself.
Parallels Between Second Language
and Mathematics Learning
Language learning and mathematics learning
are both cognitive processes. They can be
understood through a constructivist model in
which meaning is derived through the interaction
between one’s own background knowledge
and experience (top-down processing) and
one’s ability to process the given task or decode
text (bottom-up processing). These two processing
modes are complementary; one can be
used to compensate for weakness in the other,
but each alone is insufficient in terms of successful
learning and performance. They both
develop through feedback.
Top-down processing refers to higher-order
thinking skills. In mathematics, this is revealed
in conceptual understanding of structures and
patterns, appropriate application of basic arithmetic
and algebraic operations and concepts.
Similarly, in language learning, this translates
as the activation and application of appropriate
background knowledge to the creation of
meaning. However, students capable of this level
of understanding still need to rely on their
bottom-up processing skills to ensure that their
hypotheses are grounded in the details of the
text.
Bottom-up processing refers to lower level
skills that must be practiced if automaticity is
to be achieved. In mathematics, as in language
acquisition, this is associated with rote learning:
computational skills, memorizing facts and
formulas, doing mechanical, de-contextualized
arithmetic or grammar drills, and focusing on
2 • In Transit

discrete elements often without the benefit of
comprehension. Students who remain at this
level of processing cannot be said to be thinking
mathematically or using language for
authentic communicative purposes.
This view of learning is process rather than
product-oriented. It implies an explicit focus on
learning strategy use and self-monitoring if learning
competence is to be transferable. It implies
actively listening to students before attempting
to teach, taking into account their schema knowledge
and expectations of how things work. It is
incumbent on teachers to elicit and interact with
students’ prior understandings, to mediate their
difficulties so that a restructuring of students’
knowledge may take place.
The Common Methodological Debate
In both ESL and Mathematics classrooms,
deductive teaching clearly prevails. Students
are repeatedly taught the rote skills of computation
or grammar, despite disappointing
results. The debate over the merits of a deductive
versus an inductive approach goes on.
Should teachers begin by building bottom-up
skills in the hope that top-down, conceptual,
procedural understanding will somehow magically
follow? Or should they begin by ensuring
the development of top-down processing while
supporting the development of the necessary
bottom-up skills?
Uncommon Solutions
Trends toward content and task-based methodology
in ESL indicate a call for a departure from
traditional deductive teaching. In mathematics,
a statement in the Principles and Standards
document published by the National Council
of Teachers of Mathematics in 2000 claims,
“Solving problems is not only a goal of learning
mathematics but also a major means of
doing so” (Van de Walle 40). We argue that
inductive, problem-based approaches provide
a more authentic context for learning that is
not separated from doing. These approaches
are more learner-centered and engaging and
provide the window into students’ math and
language hypotheses that teachers need in
order to skillfully determine the sources of student
error. Teachers can then provide feedback
at that crucial moment when a meaningful revision
of hypotheses is apt to occur. We demonstrated
this during our workshops with two
problem-solving activities used to initiate lessons,
one from an ESL grammar book (Badalamenti
and Henner Stanchina 98) and one
reported by Ball and Bass (91–94).
By asking ESL students to study a picture of
an apartment, decide who lived there, and justify
their conclusion, the teacher was able to
elicit numerous sentences showing a range of
hypotheses on the structure: There is / there
are, including “in the bedroom has dress,” “is
a dress,” “there are have computer,” “there is
the computer,” and “there is a coffee pot on the
stove.” Similarly, by presenting the following
problem: “Joshua ate 16 peas on Monday and
32 peas on Tuesday. How many more peas did
he eat on Tuesday than he did on Monday?”
(Ball and Bass 91), the teacher elicited six different
problem-solving methods. The challenge
for both the ESL and the Mathematics teacher
lies in creating the task, identifying and interpreting
student errors, using them to understand
student learning, and making decisions
about teaching. The advantage of this methodology
for the learner is a more connected, deeper
understanding.
This approach also lends itself to a focus on
writing to learn. The Principles and Standards
for School Mathematics support communication
as an “essential part of mathematics and
mathematics education” (NCTM 60). The standards
encourage language use, particularly for
second-language learners, in order to facilitate
“communicating to learn mathematics and
learning to communicate mathematically”
(NCTM 60).
There are many ways teachers can help students
integrate language development with the
learning of mathematics. These include having
students paraphrase word problems, create
word problems, justify solutions, and explain
procedures for problem solving.
We plan to pursue our conversation about
the relationship between language and math-
Mathematics and ESL • 3

ematics learning. In particular, we would like
to focus on the level of preparedness of mathematics
teachers working with ELL students.
Within this group, there are some teachers who
are themselves non-native speakers of English,
some who may have limited mathematical
knowledge or pedagogical repertoires, and
some whose underlying assumptions and
beliefs about teaching and student failure may
not promote learning. We hope, as well, to be
able to continue our conversation with them
to seek further uncommon solutions to the very
difficult problems we face.
Works Cited
Badalamenti, Victoria, and Carolyn Henner Stanchina. Grammar Dimensions Book I Platinum Edition.
Boston: Thomson Heinle, 2000. 98–99.
Ball, Deborah L., and Hyman Bass. “Interweaving Content and Pedagogy in Teaching and Learning to
Teach: Knowing and Using Mathematics.” Multiple Perspectives on Mathematics Teaching and
Learning. Ed. Jo Boaler. Westport: Ablex, 2000. 83–104.
Chamot, Anna Uhl, and Michael J. O’Malley. The CALLA Handbook: Implementing the Cognitive Academic
Language Learning Approach. New York: Addison-Wesley, 1994. 222–250.
Dale, Theresa Corasaniti, and Gilberto J. Cuevas. “Integrating Mathematics and Language Learning.”
The Multicultural Classroom. Ed. Patricia A. Richard-Amato and Marguerite Ann Snow. New
York: Longman, 1992. 330–348.
“Division of English Language Learners.” NYC Department of Education. NYC Department of Education.
2004 .
Mestre, Jose P. “The Role of Language Comprehension in Mathematics and Problem Solving.” Linguistic
and Cultural Influences on Learning Mathematics. Ed. Rodney R. Cocking and Jose P. Mestre. New
Jersey: Lawrence Erlbaum Associates, 1988. 201–218.
“New York Rate of LEP Growth 1993/1994–2003/2004.” National Clearinghouse for English Language
Acquisition and Language Instruction Educational Programs. Office of English Language Acquisition,
Language Enhancement, and Academic Achievement for Limited English Proficient Students.
November 2004
.
“Position Statement: Mathematics for Second Language Learners.” National Council of Teachers of
Mathematics. National Council of Teachers of Mathematics. July 1998
.
Van de Walle, John. Elementary and Middle School Mathematics: Teaching Developmentally. 4th ed. New
York: Longman, 2001.
4 • In Transit