 |
What Discussions Teach Us About Mathematical Understanding: Exploring and Assessing Students� Mathematical Work in Classrooms
| |
| | Unformatted Document Text:
What Discussions Teach Us About Mathematical Understanding:
Exploring and Assessing Students’ Mathematical Work in Classrooms
Nick Fiori and Jo Boaler
Stanford University
“What is the oldest problem of pedagogy? The appearance of learning, or ‘illusory understanding’, that is, the problem of people who appear to know something that they really don't know.” — Lee Shulman (2000, p. 131)
The motivation for this paper comes from the great rewards we—as teachers and
researchers—experience every time we listen to students’ discussions about mathematics. Listening to students talk about mathematics reveals aspects of their understandings and dispositions towards mathematics that written work alone does not disclose. In particular, student discussions give us important insights into the students’ relationships with mathematics. These relationships include mathematical understanding, agency, and conceptions of the nature of mathematics. Such knowledge is crucial for assessing individuals in the classroom, and can be used to help meet goals of effective, equitable teaching.
In this study we examined videotapes of 40 groups of 3-4 students working on an open-ended
mathematical task for 90 minutes. Using first a broad lens of ‘connoisseurship’ (Eisner, 1985) we took careful note of what was revealed about students’ relationships with mathematics. For each group, we compared this knowledge to what we learned from the written work students completed in response to the task. Using categories from the New Basics Project of Australia (Department of Education and the Arts, State of Queensland, 2001) as a guide, we developed a set of categories of relationships with mathematics that are particularly conspicuous in discussions. From these categories, we developed a tool to facilitate teachers in learning from student discussions. Although our tool is inspired and influenced by recently developed fine-grained research tools for analyzing student discussions (Sfard and Kieran, 2001; Barron, 2003), our tool is designed for teachers to use, in real-time, as they negotiate the busy classroom. Once our tool was refined we applied it to the videotaped discussions. We compared the results to the written work of the students, and found that the analysis of the discussions delivered a more accurate representation of students’ relationships with mathematics.
Methodological Frameworks:
Recent work has demonstrated that researchers can obtain critical knowledge of student
understandings and dispositions by performing fine-grained analyses of students working collaboratively in small groups. Barron’s (2003) coding scheme for analyzing interactions among group members helps isolate many powerful indicators of successful student work. For example, Barron’s discovery that partner responsiveness is more powerful for predicting successful student work than prior achievement or accuracy of student ideas, is a rich and valuable finding. Sfard develops a powerful analytic tool (Sfard and Kieran, 2001) for analyzing student discourse during collaborative work. Like Barron’s analytic method, Sfard’s tool focuses on how students respond to one another’s statements. Sfard and Kieran (2001) and Kieran (2001) use this tool to carefully monitor the development of mathematical thinking in pairs of students, and to further understand how a student makes use of a partner’s knowledge.
|
| | Authors: Fiori, Nick., Jo, Boaler., Cleare, Nikki., DiBrienza, Jennifer., Sengupta, Tesha. and Shahan, Emily. |
|
| |
|
|
What Discussions Teach Us About Mathematical Understanding:
Exploring and Assessing Students’ Mathematical Work in Classrooms
Nick Fiori and Jo Boaler
Stanford University
“What is the oldest problem of pedagogy? The appearance of learning, or ‘illusory
understanding’, that is, the problem of people who appear to know something that they really
don't know.” — Lee Shulman (2000, p. 131)
The motivation for this paper comes from the great rewards we—as teachers and
researchers—experience every time we listen to students’ discussions about mathematics.
Listening to students talk about mathematics reveals aspects of their understandings and
dispositions towards mathematics that written work alone does not disclose. In particular,
student discussions give us important insights into the students’ relationships with mathematics.
These relationships include mathematical understanding, agency, and conceptions of the nature
of mathematics. Such knowledge is crucial for assessing individuals in the classroom, and can
be used to help meet goals of effective, equitable teaching.
In this study we examined videotapes of 40 groups of 3-4 students working on an open-ended
mathematical task for 90 minutes. Using first a broad lens of ‘connoisseurship’ (Eisner, 1985)
we took careful note of what was revealed about students’ relationships with mathematics. For
each group, we compared this knowledge to what we learned from the written work students
completed in response to the task. Using categories from the New Basics Project of Australia
(Department of Education and the Arts, State of Queensland, 2001) as a guide, we developed a
set of categories of relationships with mathematics that are particularly conspicuous in
discussions. From these categories, we developed a tool to facilitate teachers in learning from
student discussions. Although our tool is inspired and influenced by recently developed fine-
grained research tools for analyzing student discussions (Sfard and Kieran, 2001; Barron, 2003),
our tool is designed for teachers to use, in real-time, as they negotiate the busy classroom. Once
our tool was refined we applied it to the videotaped discussions. We compared the results to the
written work of the students, and found that the analysis of the discussions delivered a more
accurate representation of students’ relationships with mathematics.
Methodological Frameworks:
Recent work has demonstrated that researchers can obtain critical knowledge of student
understandings and dispositions by performing fine-grained analyses of students working
collaboratively in small groups. Barron’s (2003) coding scheme for analyzing interactions
among group members helps isolate many powerful indicators of successful student work. For
example, Barron’s discovery that partner responsiveness is more powerful for predicting
successful student work than prior achievement or accuracy of student ideas, is a rich and
valuable finding. Sfard develops a powerful analytic tool (Sfard and Kieran, 2001) for analyzing
student discourse during collaborative work. Like Barron’s analytic method, Sfard’s tool focuses
on how students respond to one another’s statements. Sfard and Kieran (2001) and Kieran
(2001) use this tool to carefully monitor the development of mathematical thinking in pairs of
students, and to further understand how a student makes use of a partner’s knowledge.
|
|
 Convention | | All Academic Convention is the premier solution for your association's abstract management solutions needs. | | Submission - Custom fields, multiple submission types, tracks, audio visual, multiple upload formats, automatic conversion to pdf. | | Review - Peer Review, Bulk reviewer assignment, bulk emails, ranking, z-score statistics, and multiple worksheets! | | Reports - Many standard and custom reports generated while you wait. Print programs with participant indexes, event grids, and more! | | Scheduling - Flexible and convenient grid scheduling within rooms and buildings. Conflict checking and advanced filtering. | | Communication - Bulk email tools to help your administrators send reminders and responses. Use form letters, a message center, and much more! | | Management - Search tools, duplicate people management, editing tools, submission transfers, many tools to manage a variety of conference management headaches! | | Click here for more information. |
|
|
|
| |
|
|
|
Page 1 of 9 
