Learning and Mathematics

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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.

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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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