Talking Mathematics with the Teacher while
Working in Small Groups in an Algebra I Class:
Some Differences and Similarities in Discourse for Stronger and Weaker
Students
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Judith Kysh University of California, Davis jmkysh@ucdavis.edu |
The
purpose of this study was to better understand what and how students learn
through their talk by examining the discourse between teacher and student as
students worked on problems in small
groups in a diverse Algebra 1 class.
This summary report is an overview of group results based on data
gathered in relation to the questions:
How do students differ from each other in their talk about their mathematical
work? and How does the teacher respond to differences in students' talk about
their mathematical work? The full report
includes further analyses of individual transcripts and provides more detailed
discussion of differences and similarities among students.
The national reform effort in mathematics education calls for a constructivist approach to teaching mathematics. Recently developed curricula, including the program used in this study, are based on the theory that students construct their own understanding of mathematics and that teachers and materials can be prepared to better serve students in helping them to develop this understanding. These materials generally include problems designed for small group work so students can talk about their work as they are doing it.
Articles written in the mid-nineties urged researchers to study classrooms in which teachers were attempting to help students develop their own understanding through the social negotiation of meaning as they worked together and talked with each other and the teacher (Cobb, 1996, Lerner 1996), and there is still much to be learned in relation to language use in mathematics classrooms particularly high school. Few researchers in mathematics education work as regular teachers in classrooms and report on their attempts to work with diverse groups of children in that setting. Lampert (1985), Ball (1996), Parker (1993) and Romagno (1994) have been exceptions. Lampert at the middle school and Ball in a third grade classroom have focused their work on generating thoughtful whole group discussions. Parker at the fifth grade level and Romagno in a ninth grade basic mathematics class, team taught with the regular teacher and focused on developing alternative curricula and teaching methods to engage students in thinking more deeply about mathematics. There are not long term studies of high school classrooms in which the researcher is the teacher, nor are there year-long studies focused on the discourse that occurs between students and teacher as the teacher circulates to work with the students as they work on problems in their small groups.
Methods:
Two questions guided this study. How do students differ from each other in their talk about their mathematical work? How does the teacher respond to differences in students' talk about their mathematical work? Because these were two of seven questions about what happens in a typical classroom when students work in small groups, I used an ethnographic approach. I arranged to teach an Algebra 1 class from October to June, in an inner-city school with a mixed population of Asian, African-American, Latino, and White students. The class was using Mathematics 1 materials, a replacement course for Algebra 1 designed to enhance students' problem solving, reasoning, and communication skills and to use the other areas of mathematics, geometry, graphing and functions, probability and statistics, as a basis for understanding algebra.
As the teacher, fully responsible for all aspects of the class including attendance, grading, and talking with parents, I could be an insider, but because I was only teaching one class I would still be an outsider in the some important respects. While I did not live through a regular teacher's long and demanding five-period teaching day, my other full time responsibilities did not allow me any more preparation time for teaching than a regular teacher would have for such a class, so it was easy to stay with my plan of using the materials as recommended and not supplementing or changing them significantly. Using the materials as they were written was important to my goal of working in a classroom situation that might be considered close to a normal class where I would experience many of the same pressures and dilemmas as a regular teacher (Ball and Lampert ).
Data:
I gathered data in the four categories described by Eisenhart (1988): participant observation, interviews, collection of artifacts, and reflections including "emergent interpretations, insights, feelings, and the reactive effects that occur as the work proceeds." (p.106) To supplement my observations as a participant, I used an audio-tape recorder. To record teacher-student exchanges, I wore a small tape recorder and an external microphone as I moved around the classroom from group to group. Artifacts gathered included written work and records: all the assignments the students did throughout the year, my gradebook records, their mathematics grades in previous and following courses, and information from the counseling office.
The class was composed of 33 students who were in grades 9-12. The larger study focused on the 21 ninth and tenth graders. The analysis in relation to change in discourse is based on a chronological sequence of transcripts for each of the eleven ninth and tenth grade students, who were in the class from November through May and for whom I had both early and late recordings on the 28 classroom tapes that were transcribed. Analysis was based on on-task talk, which meant the exchange was related to the mathematics we were working on. On-task turns were categorized as About, In, With, or Beyond mathematics based on an idea from Brenner's work (1995).
Another part of the analysis included identifying topically related sets, TRS's (Cazden, 1988) in the on-task talk and looking for discourse patterns in those sets. In considering differences among students, I examined the initial questions they used to open the dialogue and categorized opening turns as requests for help, requests for verification, or requests for clarification. I also categorized degrees and forms of closure.
At the end of the school year, before transcribing all of the tapes, I ranked the 21 students from strongest to weakest (numbered 1-21) based on my holistic assessment of their level of understanding of algebra, problem solving, their facility with algebraic operations, and the likelihood of success in Geometry and Algebra II. As I considered the use of language In, With, or Beyond, mathematics I realized that much of the language In mathematics used by the weaker students appeared in one to four word phrases. I then re-examined a sample of the transcripts of all the students to classify on-task turns as questions, phrases, or statements.
As I compared TRS's three forms of teacher-response emerged. Level 1 involved questions or prompts to get started, but then leaving the work to the student. Level 2 included explanation and direct instruction and usually meant longer turns for the teacher. At Level 3 the teacher became involved in working through the problem with the student.
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Results: As might be expected stronger students required less closure and posed a much higher percentage of initial questions that asked for verification, while weaker students were more likely to ask for help. Figure 1 shows the relationship between class rank (1-16) and percentage of initial questions that were for verification (r = -0.57). Only 16 students are included because the other five almost never initiated a TRS. On the scatter plot Hoang and Sam are not in line with the others, and Hoang is clearly an |
Figure 1 |
outlier. Without Hoang the correlation is r = –0.76 and without Hoang and Sam r = –0.89. As an able problem solver, Sam never hesitated to ask for help, unlike most of the other good problem solvers who were not so sure of themselves. Hoang, on the other hand, was not as strong a problem solver as his class rank might imply. He was an extremely hard-working and thorough student who asked about everything he was unsure of.
The results for closure showed a strong correlation between the need for more closure and class rank (r = 0.70). Again Hoang was an outlier, and without Hoang the correlation increased to 0.84.
More interesting, I think, is the correlation between class rank and percentage of responses that were statements. Stronger students more often completed their sentences while weaker students responded with one or two words, or phrases. I discovered this pattern when comparing November and May transcripts for the highest and lowest ranked students. The table shows the number and percent of turns classified as statements, responses, or questions for two early TRS's for Sam and Jonathan and for Carolina and Tsaan Fou. Sam, a confident problem solver, consistently made statements, without being overly concerned as to whether they were right or wrong. He gained information about the solution of a problem either way. Jonathon, also a strong student and an independent thinker made statements. Tsaan Fou, one of the weaker students, like Carolina gave short responses that were not sentences. While one reason for Carolina's short responses might have been that she was still learning English and was more comfortable speaking Spanish, Tsaan Fou was bilingual and spoke only English in school. Both lacked confidence in their ability to solve most of the problems they asked about.
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Number |
Percent |
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Statement |
Response |
Question |
Statement |
Response |
Question |
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Sam |
7 |
1 |
3 |
64 |
9 |
27 |
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Jonathon |
4 |
1 |
2 |
57 |
14 |
29 |
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Carolina |
9 |
36 |
11 |
16 |
64 |
20 |
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Tsaan Fou |
0 |
7 |
3 |
0 |
70 |
30 |
Table 1
Student Responses
When I compared transcripts from early in the year with those recorded in May, I found an increase in percentage of statements and length of responses for both Carolina and Tsaan Fou. There was no change for Sam or Jonathon. The fact that stronger students used a greater percentage of complete sentences means that they took more opportunities to articulate mathematical relationships, a practice that could be very useful in terms of both understanding and remembering those relationships. A goal for teachers might be to focus on tailoring their questions to elicit full sentence responses from all students and to avoid settling for phrases.
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Surprisingly there was no discernible difference in percentages of language use In or With mathematics between stronger and weaker students. Figure 2 is a scatter plot comparing class rank and percentage of all turns (not just initial questions) In, With, and Beyond mathematics ( r = –0.097 ). Arranging students in order from highest to lowest percentage of use of language In, With and Beyond mathematics led to some further observations about gender differences |
Figure 2 |
and English language learners. The list shows percentages largest to smallest with the girls percentages underlined. The Spanish speaker's percentages are in bold and Chinese, Mien, and Vietnamese speaker's percentages are in italics. Percentages for those who spoke only English are in regular type.
70, 68, 60, 57, 55, 55, 53, 53, 52, 52, 52, 48, 47, 45, 42, 41, 39, 32, 26
The boys are at one extreme or the other, while the girls cluster around the median. The Spanish speakers show no clear pattern while the Asian language speakers tended to use a lower percentage of mathematical language. While the transcripts for each individual provided reasons for his or her position in this hierarchy of language use, generalizations were not possible.
For Level 1 and Level 3 teacher responses, comparisons of TRS's for strong and weak students showed that the type of response seemed to depend on the type of problem or question posed more than on who posed it. While weaker students elicited more Level 3 responses, stronger students elicited their share, and comparisons of TRS's line-by-line, for strong and weak students showed similar discourse patterns. Level 2 responses, on the other hand, were regularly elicited by some students and not by others based on the teacher's assumptions about the student's expectations. Students most likely to receive Level 2 responses were three students who were reluctant to respond and two who were in their "silent phase" of learning English where they could understand but lacked confidence in their ability to speak. My response to their reluctance to respond, in either case, was often a too lengthy explanation, and this seemed to become my habit in talking with these students throughout the year.
The results of this study, many of which are in the details of examining each student's talk, have implications for teachers' expectations in relation to working with students in small groups particularly in how we pose our questions and our expectations for their responses. Clearly further examination of student discourse is needed. This study is just a small part of beginning.
References:
Ball, D. & Wilson, S., (1996). Recognizing the fusion of the moral and intellectual. American Educational Research Journal 33(1), 155-192.
Brenner, M. (1995) "Development of mathematical communication in algebra problem solving groups: focus on language minority students." Research report, Linguistic Minority Research Institute, University of California, Santa Barbara.
Cazden, C. (1988). Classroom discourse: The language of teaching and learning. Portsmouth, NH: Heinemann Press.
Cobb, P. (1996). Where is the mind? A coordination of socio-cultural and cognitive constructivist perspectives. In C. Twomey Fosnot, (Ed.), Constructivism, theory, perspectives, and practice. New York, Teachers College Press.
Eisenhart, M., (1988). The ethnographic research tradition and mathematics education research. Journal for Research in mathematics Education , 19(2), 99-114.
Lampert, M., (1985). How do teachers manage to teach? Perspectives on problems in practice. Harvard Educational Review. 55(2), 178-194.
Parker, R. (1993). Mathematical power: Lessons from a classroom. Portsmouth, NH. Heinemann Press.
Romagno, L. (1994). Wrestling with change: The dilemmas of teaching real mathematics. Portsmouth, NH. Heinemann Press.