Comments from package users (A)
Comments on using the package to enhance a math content course
for elementary pre-service teachers (by Larry Feldman & Margaret
Stempien)
The following comments are provided by Larry Feldman and Margaret Stempien, two mathematics educators on the faculty of Indiana University in Pennsylvania who teach math content and math methods courses to pre-service teachers in both secondary mathematics and elementary education. These mathematics education professors reviewed these materials together, and then used them to re-design Professor Feldman's math content course.
We highly recommend the use of these materials for anyone who is considering seriously implementing the spirit of the current reform movement in the teaching of mathematics. While so many materials put a transparent layer of reform on their old editions, these materials have the potential to really transform curriculum, instruction, and assessment.
These materials are filled with ideas that can be integrated into mathematics teaching from Kindergarten through graduate school. They give videotaped examples of the materials being used by sixth graders, pre-service and in-service teachers, and by mathematics educators. The key concept is that mathematics can and should be taught with an investigation approach, where a unifying theme is at the heart of instruction, not the mathematical subtopic found in the next section of the textbook.
The two of us as mathematics educators found ourselves stopping the tapes many times to discuss both exciting mathematical issues and issues of teaching. We found the materials to be empowering for our own teaching.
For example, Larry has changed the structure of a course on geometry, probability, and statistics that is required for all elementary education undergraduates at Indiana University of Pennsylvania. The semester is now organized around three investigations -- one on fair games (for probability), one on doing a statistical study, and one on geometric shapes (including tessellations). Small groups work together on these investigations and give reports of their findings to the class as a whole.
Larry had taught this course with a great deal of hands-on activities and a focus on teaching for understanding of underlying concepts rather than on memorization of facts. However, after going through these materials he felt comfortable in going a further step. The textbook used for the course claimed to have been updated to reflect changes suggested by the NCTM Standards. However, the text and the traditional focus encouraged a relatively passive role for the student. The concept to be taught on a given day was usually quite fixed and the "hands-on" approach required students to replicate the "correct" procedure.
Under the investigation approach now being used, students work in groups of (mostly) four students on three investigations over the entire semester. There is still a focus on teaching for understanding and on the use of hands-on materials but there is now a unifying theme for each investigation and a more active role for students.
The Borasi / Fonzi materials have laid a framework for the investigation approach in this class for pre-service elementary teachers. Many of the specific ideas have been borrowed from this work, such as ideas for investigating attributes of geometric shapes and of tessellations. We will also be using these materials in other courses, such as Margaret's expansion of some of these ideas in a Mathematics 101 survey course. We will also be referring to this approach in graduate courses we teach in Curriculum and Instruction (one for Elementary / Middle School Mathematics and one for Secondary Mathematics).
More important than the specific ideas for improving classroom practice is the encouragement for shifting one's philosophical paradigm. It takes some courage to use the textbook essentially as a reference or occasionally for part of an assignment. If an instructor does not have the confidence to make the bold moves suggested by these materials, success will be unlikely. These materials can provide a solid background to this approach.
Standard curriculum materials try to show mathematics with clear definitions, direct expositions of topics that neatly flow from one to another, and exercises that directly match the topics explained. However, real mathematical development is "messy". By starting with a rich situation as these materials suggest, students can explore what it is like to actually do mathematics at a level appropriate to the student. If the goal is for sixth-grade students to explore the world of tessellations (one investigation from these materials), not only do solutions become "messy" but also so does the nature of the questions to be posed. This "messiness" can lead to much higher-level thinking than having students process the final products of the "clean" textbook. For example, in these materials students frequently are asked to come up with their own definitions as opposed to the traditional sequence that starts with definitions.
We hear the U.S. mathematics curriculum described as "a mile wide and an inch deep". Students learn many isolated facts without being challenged to develop their own questions and methods for solutions. At first glance, one might think that an investigation approach would involve cutting away many important topics, yet most if not all of the key topics can be integrated into thematic units such as these.