Language and Mathematics - Cocking and Chipman (1989)
Cocking and Chipman examine the mathematical ability of language minority -- particularly bilingual -- students, attempting to identify linguistic and
cultural variables that might explain why their mathematical ability falls
increasingly behind that of students who speak English as their primary
language ("majority students").
First Cocking and Chipman investigate the relation between language and
math ability; then they look at external influences on performance such as
teacher competencies and attitudes and parental attitudes and support. The
focus is primarily on Hispanic students, with some support from data on
Native Americans.
Chapter:
Cocking, R. R., and Chipman, S. "Conceptual Issues Related to Mathematics
Achievement of Language Minority Children". In Cocking, R. R., and Mestre,
J.P. Linguistic and Cultural Influences on Learning Mathematics.
Hillsdale: Erlbaum. 1988, pp. 17-46.
Overview (and some applications)
- Several factors influence student performance in mathematics as well as
in other subjects: entry-level knowledge, opportunities to learn at school
and at home, and motivation, including cultural and parental attitudes
towards math. Cocking and Chipman cite Hispanic women and Native Americans
who perceive mathematicians as sloppy, remote, obsessive, and calculating
(as in scheming, not adding up numbers!) and thus tend to shy away from
mathematics.
[These factors raise the issue of perception of and real understanding
about what mathematics is and who mathematicians are. One implication of
Cocking and Chipman's data is that it might be useful to ask students to
talk about their feelings about mathematics. Follow up questions would
permit discussion of difficulties as well as what does work, and would
certainly provide the teacher with more information about why the students
think or feel the way they do.
The benefit of direct questionning is that the student feels heard and the
teacher has a chance to respond. Such direct questionning also suggests
where the real starting point for learning in the class needs to be. Note,
of
course, that the student doesn't need to love math to do math. What the
teacher likes about math could be shared, however, and the kinds of things
that might help the student learn math could certainly be discussed.
Obviously, while direct questions might be optimal in some settings,
anonymous questionnaires could also be used to assess the same information.
- In general, apart from bilingualism, positive correlations have been
found between math and verbal abilities, although some researchers raise
the question whether these result from a dependence of mathematics on
language, or because both involve general intellectual skills. In any
case, the nature of the relation between the two remains uncertain.
[In the classroom, as previous postings of summaries of articles by
Collins, Brown, and Newman (Apprenticeship); Brown, Campione, Reeve Ferrara
& Palincsar (Interactive Learning); Schoenfeld (Metacognition); Resnick
(Mathematics as an Ill-Structured Discipline); and Lampert (Knowing, Doing
and Teaching Multiplication) all suggest, talking about math enhances the
understanding of math. Thus, were students stronger in verbal abilities,
their understanding of the mathematics being taught would be enhanced by
working together with a partner to solve a problem, do a quiz, etc.
Depending on the specific skills being developed in the problem, it might
also be useful for the student to be paired with a student of like ability
(rather than pairing a strong student with a weaker student), so that the
students would really share a language for discussing the problem(s). Such
an arrangement in the classroom would mean that more able students would be
able to move rapidly in their conceptual understanding, which would allow
the teacher time to work in a more focused way with weaker students.]
- Cocking and Chipman found that students had the most difficulty in
translating words into mathematical symbols. For example, the sentence
"There are twice as many students as desks" was often expressed as "D=2S"
rather than "S=2D". Although this conceptual error occurred whether or not
the problem was given in the student's primary language, it signalled an
extra level of difficulty for language minority students because of the
additional translation required. Cocking and Chipman suggest that the
remedy for this is increased exposure to math. Experience, they suggest,
may be the most important factor in predicting achievement in mathematics.
[There are numerous ways to increase students' conceptual understanding of
the content of mathematics. Obvious possibilities for the teaching of
Geometry include use of The Geometer's Sketchpad or Cabri Geometre. In
addition, however, Cocking and Chipman point out conceptual difficulties in
using language to describe the mathematics being undertaken. Some more
specific, in-class kinds of examples to address this issue might be asking
students to write a problem and find an answer to it--students may create
problems that are easy to solve; however, research suggests that once they
feel comfortable they are more likely to pose difficult problems to
challenge themselves.
In fact, a student's range of comfort may be beyond what the teacher
expects, and even within one's comfort level creating problems for oneself
may prove more interesting and challenging than tackling those the teacher
would normally assign. Such student-generated problems could become
full-class exercises; students in a class might solve each others'
problems, using those who generated given problems as resources.
The benefit of these kinds of exercises is that they provide connections
between students' understanding of a problem and the same material as a
mathematical problem. The process of explaining and reexplaining what to
do and how it's done enables students to firm up their understandings--and
it is fun!]
- Teachers' attitudes are almost as important as those of students.
Cocking and Chipman report that minority teachers may themselves express
negative attitudes about math, encouraging their students to pursue higher
levels of education but not in math or science. Majority teachers may tend
to shield minority students from failure by holding lower expectations and
not recommending higher-level math classes.
[It's interesting to note that success stories in inner city schools have
for the most part evolved out of teaching situations that truly challenged
students, not those which shielded, protected, or pandered to them.]
Direct Quotes (and more comments)
"Math achievement is heavily dependent upon school instruction..., and
it is not likely that math achievement would be related strongly to family
background variables tied to socioeconomic status. Occupational
expectations and information associated with socioeconomic status may
affect the value assigned to mathematics study and achievement. The family
background variables...likely to affect...mathematics are such things as
family member attitudes toward mathematics and mathematicians, and the
early experiences the child has in environments that convey these
attitudes" (p. 32).
"[T]he most fundamental mathematical concepts are equally present in
children from disadvantaged backgrounds. However, even at the preschool
level, there are differences in communicative competence that presage later
measured differences in school achievement. More attention needs to be
given to the organismic variables--such as development--and to the
environmental variables--such as home and school opportunities for
learning--that determine what kind of mathematical competence is ultimately
built upon these shared conceptual foundations" (p. 23).
[One example of this is parental assistance, which tends to taper off as
topics exceed parents' knowledge, contributing to a decline in math
achievement at the higher levels.
Another approach to working with students to develop a conceptual
understanding of the problems they are assigned might include very
structured group work in solving problems. Students could first be
assigned to groups of four; next, the teacher would direct each student to
assume responsibility for a different role--articulating the question being
addressed, showing how to address it, working it through, and stating what
the answer means in terms of the question asked) for each of four problems.
This approach would give all the students a clearer understanding of the
various steps in solving the problem, and would provide support if one of
them got stuck.]
-- Summarized by Andrea Hall
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Learning and Mathematics
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