Comments from package users (A)
Comments on using the package as a text in a seminar for doctoral students
in mathematics education (by Beatriz D'Ambrosio)
The following comments are provided by Beatriz D'Ambrosio, a faculty member at Indiana University/Purdue University at Indianapolis responsible for teaching several mathematics education courses at both undergraduate and graduate level. Professor D'Ambrosio is also involved in the design and delivery of professional development for in-service mathematics teachers in her role as Principal Investigator in an NSF-funded "Local Systemic Change Project." Professor D'Ambrosio found several possible ways in which the materials in this package could enhance her practice as a teacher educator -- including using the materials to enhance her math content and math methods courses for elementary pre-service teachers, to design some professional development experiences for in-service teachers, and even assigning them as a reading within a seminar for doctoral students in mathematics education.
In this commentary, she reports on how she used the package as one of the texts in a doctoral seminar on "Constructivism in mathematics education," as a means to invite her students to explore implications of constructivism for mathematics teacher education.
As I reviewed the materials by Borasi and Fonzi (1998) it became apparent that I would find many uses for the package. The most immediate use came in planning a doctoral seminar on "Constructivism in Mathematics Education", since I was planning and reviewing the package simultaneously. With permission from the authors, I incorporated the package as one of the texts (still in draft form) to be used in the seminar.
My goal for the seminar was to help doctoral candidates build an understanding of student's construction of mathematical ideas and an image of the teaching practices that would support the construction of knowledge. In particular I wanted the candidates to explore the potential of an inquiry approach to mathematics teaching. I was also concerned that the doctoral students explore their future roles as teacher educators in programs that purport to graduate teachers who teach within a constructivist framework.
The package offered my students the following components: essays that elaborated on the theoretical foundation for an inquiry approach to teaching mathematics, guidelines for the professional development of teachers (and pre-service teachers) with the goal of helping them understand an inquiry approach to teaching, and most importantly, powerful images that captured both an inquiry-based classroom with children and an inquiry-based professional development experience for teachers (through the video-tapes).
In order to encourage students to carefully explore the entire package I structured an assignment asking them to review the materials and prepare a critique that could be sent to the authors as part of the "pre-publication" review.
A second assignment was to build a mathematics content or mathematics methods component for a teacher education program that would work towards the professional development of future teachers to understand an inquiry approach to teaching. As might be expected, several students drew on the "supports" offered by the package in order to conceptualize the content or methods component of the teacher education program they were designing. In one case the doctoral student (also a graduate assistant on an NSF-funded project) planned a full day of professional development for middle school math teachers drawing heavily on the materials from the package.
One of the most striking aspects of the package, was its ability to cater to the learning needs of doctoral students who were at many different levels of understanding of constructivism and inquiry-based teaching. For the most novice doctoral students the package was instrumental in helping shape their images of an inquiry-based approach to teaching. The video-tapes of professional development experiences and the journal entries of teachers afforded students the opportunity to examine and reflect on many of the questions raised by participating teachers. This process pushed their thinking about inquiry to a deeper level than merely reading the essays might have succeeded in doing. Questions about the reluctance of participants to initially accept an inquiry approach were also raised and discussed as a result of reading the many components of the package, giving doctoral students insights into the complexities of professional development.
For the most advanced doctoral students, the materials were instrumental in helping them conceptualize the design of teaching experiences, for both professional development opportunities as well as pre-service teaching opportunities. Students incorporated many aspects of the framework in the planning of their courses for undergraduate pre-service teachers. The flexibility of the materials and the many different dimensions and components available (from theoretical readings to actual lesson-plans and timelines) allowed students to meet their individual needs, as they sought to understand inquiry-based teaching of mathematics.
Students raised and examined philosophical questions about the role of definitions in mathematics, motivated by the activities of the tessellation illustrative unit. They explored questions of the fallibility of mathematical ideas, the humanistic nature of mathematical developments throughout history, the role of errors in the construction of mathematical knowledge and other issues directly linked to aspects of the construction of knowledge that emerge as one explores the materials available in this package. We explored the implications of these issues for the teaching of mathematics and examined the artful ways in which Borasi, Fonzi and the other instructors teaching in the videos address, with teachers, the complex nature of students' construction of mathematical knowledge.
The materials provided a theoretical framework about inquiry teaching, provided examples of inquiry instruction, provided examples of challenging mathematical tasks, and raised many questions and issues for discussion. They served as a tapestry of materials that provided a backdrop for a conversation about the many elusive concepts involved in understanding the construction of mathematical knowledge through inquiry-based teaching.