During the first year of a district-wide K–8 mathematics initiative, teachers participated in a week-long summer institute where they were introduced to the district's newly-adopted mathematics instructional materials. Part of this summer orientation included an overview of the scope and sequence of the mathematics curriculum, a demonstration of the spiraling concept of the instructional materials, and an overview of the alignment of the instructional materials to national, state, and district curriculum standards.
Monthly professional development sessions during the school year were designed to have teachers further examine the organization of the mathematics content within the instructional materials (e.g., how key mathematical ideas were sequenced and developed throughout the curriculum) and how the units and activities were expected to contribute to students' learning of the content. Working in mixed grade level groups, teachers engaged in a number of the mathematical activities within the instructional materials and looked at articulation of mathematical ideas across grade levels. For example, the teachers worked backward through the grade levels examining lessons that set the stage for representing relationships by graphing Cartesian coordinates. The primary goal of this strategy was to identify the trajectory of work with algebraic representations that undergird success in coordinate graphing with understanding. The teachers came to understand that using a coordinate grid to name the location of a point in the plane was necessary but not sufficient to learning to use a point in a plane to represent the relationship between related measures. They also considered connections between bar graph representations of categorical data and Cartesian representations of continuous data. In doing so, the trajectory from topics in the elementary grades to application of Cartesian representation in the middle grades became clearly articulated for the teachers.
These professional development experiences served as the catalysts for rich discussions about the content addressed throughout the curriculum, and helped teachers identify the foundational ideas necessary for students' further learning of the mathematics.
One component of a professional development program for sixth grade science teachers was to engage them with examining the alignment of their district-adopted instructional materials with the state curriculum content standards. For example, one session focused on how the instructional materials addressed the standard "The earth processes we see today, including erosion, movement of lithospheric plates, and changes in atmospheric composition, are similar to those that occurred in the past."†
The sessions were facilitated by content experts and teachers with particular expertise in the use of the instructional materials. Participating teachers examined the instructional materials for the learning goals of specific activities, the level of cognitive demand of each activity, and the coherence of the set of activities in addressing the concepts specified in the standard. Substantial time was spent developing concept maps of the content teachers are expected to teach to address each standard. Participating teachers reported a far better understanding of their science curriculum and the strengths and limitations of the materials they are charged with using. After the analysis was completed, a subset of teachers worked with content experts to develop resources for their colleagues, providing them with information on how to use their instructional materials effectively to create high quality opportunities for students to learn the state curriculum content standards.
† National Research Council. (1996). National science education standards. Washington, DC: National Academy Press.