Transcript Teaching Math through Problem Solving

Mathematics and ESL –
Common Ground and
Uncommon Solutions
By
Prabha Betne (from Math) &
Carolyn Henner Stanchina (from ESL)
LaGuardia Community College, NY
5/7/2005
Background
We are presenting what we have learned
from our collaboration on a series of
workshops for teachers of mathematics
struggling to help English language
learners who are, in turn, struggling to pass
the Mathematics A Regents examination.
Objective
We will discuss
• The parallels between second language
and mathematics acquisition and
methodology
• The roles of teachers and the immense
challenges facing an education system
dealing with approximately 150,000
students who are classified as English
Language Learners (15-17% of total
enrollment) speaking some 140
languages.
MATH Standards
• The 1990 revised academic standards for high
school graduation requires all students, including
English Language Learners, to pass five Regents
examinations, including Mathematics, with a
score of 65 or above.
• Simultaneously, the Mathematics Regents
has shifted focus from de-contextualized
manipulation skills to contextualized
problem-solving strategies.
Our Pre-Workshop
Questionnaire Results
Analysis indicated
 Lack of collaboration between ESL and math
teachers
 Some teachers had developed strategies for
addressing the students’ overwhelming linguistic
and cultural barriers to learning
 Others blamed the students for a lack of critical
thinking skills, lack of practice in math, and their
“unwillingness to catch up in English reading
abilities.”
The Language of Math
Contextualizing math has resulted in the
use of culturally-bound concepts which
may not be part of the ELL students’
schema knowledge.
Language of math lacks redundancy,
therefore, there are few clues to meaning
and guessing.
Language of Math (Continued)
 In addition to specialized vocabulary, there are
long noun phrases, complex collocations,
confusing prepositions and cultural differences in
the symbolic denotations of mathematical
processes.
 While certain words designate specific math
operations, (“less than” indicates subtraction), the
opposite may seem true (as in: Jerry has 8 CD’s.
He has 4 less than Mike. How many does Mike
have?) (Mike has 8+4)
Language of Math (Continued)
 Students map the surface syntax of the problem
statements onto their equations. Given the
problem, “There are 5 times as many students as
professors in the math department. Write an
equation that represents this statement,” students
typically write 5S=P because they follow the
literal word order of the natural language
sentence.
This brief description of the challenges
inherent in solving mathematical word
problems mirrors the learning process itself.
Parallels between Second
Language and Math Learning
Language and mathematics learning are both
cognitive processes.
They can be understood through a
constructivist model in which meaning is
derived through a combination of one’s own
background knowledge and experience (topdown processing) and one’s ability to process
the given task or decode text (bottom-up
processing).
Parallels between Second
Language and Math Learning
(Continued)
These two modes are complementary: one can be
used to compensate for weakness in the other, but
each alone is insufficient in terms of learning and
performance. They benefit from the activation of
cognitive, meta-cognitive and affective learning
strategies. They develop through feedback on topdown and bottom-up processing.
Parallels between Second
Language and Math Learning
(Continued)
Bottom-up processing refers to lower level skills
that must be practiced in order to achieve
automaticity.
In mathematics as in language learning, this is
associated with memorizing facts and formulas,
focusing on discrete elements often without the
benefit of comprehension, doing mechanical, decontextualized arithmetic or grammar drills.
Parallels between Second
Language and Math Learning
(Continued)
Top-down processing refers to higherorder thinking skills. In math, this is
revealed in conceptual understanding of
structures and patterns, appropriate
application of basic arithmetic and
algebraic operations and concepts.
Parallels between Second
Language and Math Learning
(Continued)
Similarly, in language learning, Top-down
processing translates as the activation and
application of appropriate background
knowledge to the processing of meaning.
Both models imply learning strategy use,
self-monitoring, and learning transfer.
Implications for Learning
If the students’ initial understandings or
preconceptions are not engaged, they may fail to
grasp new information and concepts.
It is incumbent, then, on the teacher to elicit and
interact with students’ prior understandings, to
mediate their difficulties so that a restructuring of
students’ knowledge may take place.
Uncommon Solutions
Trends toward task-based methodology in
ESL, as well as a statement in 2000 in the
Principles and Standards document published
by the National Council of Teachers of
Mathematics that “Solving problems is not
only a goal of learning mathematics but also a
major means of doing so,” represent a call for
change.
Uncommon Solutions
(Continue)
The inductive, problem-based approaches provide
a more authentic context for learning which is not
separated from doing.
 They are more learner-centered and engaging
because they provide the window into students’
math and language hypotheses which teachers
need in order to skillfully determine the sources
of student error and provide feedback at that
crucial moment when a meaningful revision of
hypotheses is apt to occur.
Uncommon Solutions
(Continue)
 This approach also lends itself to a focus on
writing to learn. Paraphrasing to check one’s
comprehension of a word problem, writing to
define, explain a concept or demonstrate a
procedure for problem-solving, to create word
problems, or writing to record learning
experiences and insights in a journal, are only
some of the ways we can systematically integrate
language development and mathematics and
enhance the learning of both.
Conclusions
We plan to pursue our conversation about the
relationship between language and
mathematics learning.
In particular, we would like to focus on the
level of preparedness of mathematics
teachers working with ELL students.
Conclusions (Continue)
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 share.
Thank You