by Bob Trammel
In my school district, we have been using an integrated secondary mathematics curriculum for the past two years. The teachers are now sold on its merits, but much convincing and a considerable amount of professional development work were needed to make it successful. A fundamental reason for this groundwork is that an integrated curriculum is very different from a traditional curriculum. After looking at how the two curricula and approaches to them may differ, we can consider what is needed to make an integrated program work.
Integrated mathematics is not new mathematics but old mathematics taught and assessed in new ways. A foundation for any mathematics content exists in the NCTM's Curriculum and Evaluation Standards for School Mathematics (Reston, Va.: NCTM, 1989) and Principles and Standards for School Mathematics (Reston, Va.: NCTM, 2000). The "big ideas" in mathematics are strands woven within grade bands. In my district, the integrated curriculum at the high school level contains four strands: algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics. So what makes the content of an integrated mathematics program different? A traditional program often uses a horizontal model with first-year algebra, geometry, and second-year algebra taught in a linear fashion. In an integrated program, a vertical model that interweaves the three is common.
Also, the lesson structure in the integrated mathematics program differs from that used in a traditional program. An integrated mathematics lesson typically begins with a context-based problem, and concepts emerge as students attempt to solve the problem. In contrast, a traditional mathematics lesson typically begins with the presentation of a mathematical concept and ends with students' attempting to apply the concepts. In the integrated mathematics lesson, students immediately experience the usefulness of the mathematics.
Another potential difference between integrated and traditional mathematics is that students who are participating in an integrated program usually analyze data and make and test conjectures about mathematical models. These activities are vastly different from simply applying a model given, as in most traditional programs.
A teacher's role in an integrated mathematics classroom normally shifts to that of a facilitator who uses probing questions to stimulate students and interact with them. In such a role, the teacher guides the instruction in an inquiry-based approach. This role is in stark contrast to many direct-instruction programs in which the teacher is primarily a lecturer.
Small-group cooperative learning may be used extensively in the integrated mathematics classroom as an instructional strategy. The teacher, circulating among and interacting with the groups, encourages them to find ways to solve problems without giving them an example to be emulated, as in traditional classrooms. However, having four to seven cooperative groups simultaneously interacting requires solid classroom-management skills and careful planning by the instructor. Very few experienced teachers received any undergraduate training on managing classrooms in which students work in cooperative groups. New to this type of instructional strategy, many teachers quickly find that moving three desks together does not result in cooperative group learning. Our district found it essential to offer many in-service opportunities in the art of cooperative group dynamics for all teachers.
Technology is another integral part of most integrated mathematics curricula. Graphing calculators are frequently recommended, and downloaded software programs can be used to enhance instruction. Again, professional development workshops have been vital to success in using technology with a curriculum using real-world data. Unlike many traditional programs, many integrated mathematics programs demand the technology.
Finally, assessment models for integrated curricula are normally structured with informal and formal components. Most teachers use written journals, cooperative group evaluations, quizzes, and in-class examinations. Take-home assessment activities and extended projects are other options. A variety of assessment in-service opportunities for teachers unaccustomed to this style of testing instruments have been necessary in my district.
The bottom line is that integrated mathematics not only rearranges mathematical content but also requires a new instructional delivery system. Both may be highly desirable in traditional programs but are not deemed essential by many teachers. In our district, in-service experiences involving both content and delivery strategies for teachers have been and will continue to be essential. The success of an integrated mathematics program relies on in-service training and a continuing commitment from the superintendent to provide appropriate funding. Such a program is worthwhile for students.
|Bob Trammel is the mathematics curriculum coordinator for Fort Wayne (Indiana) Community Schools. With more than 30 years of teaching mathematics at the secondary level and several teaching awards, Bob has done professional consulting in other school districts and has worked on the Indiana standards. He gave a presentation on integrated mathematics at the NCTM Annual Meeting in Chicago in April 2000 and at the Indiana state meeting in November 2000.