In the previous paper we discussed the challenges of instructional leadership for reforming mathematics education in a K-4 school. In this paper we describe how the school principals and teachers-leaders developed authentic assessment consistent with NCTM Standards (1989, 1991, 1995) and constructivist learning theory (Cobb & Yackel, 1996; von Glasersfeld, 1995). Furthermore, we explain the kinds of challenges and dilemmas the educators encountered when a state mandated mathematics test was initiated.
Early in the reform effort, few textbooks existed that addressed the NCTM Standards recommendations. Elementary teachers and principals researched and collaborated with university and secondary mathematics educators to understand mathematics content and develop a pedagogy grounded in constructivism. Within this collaborative structure, teachers invented and restructured mathematics curricula that centered on main mathematical ideas such as: unitized systems, zero/infinity, change, chance, dimensionality, location, and the key processes of combining, comparing and partitioning. It became necessary for educators to also develop mathematics lessons aligned with "restructured" curricula that valued problem-solving, reasoning, communication, modeling, illustrating, and student experiences (NCTM Standards, 2000).
Performance Tasks and Student Solutions
To evaluate instruction and student understanding of these mathematical skills and ideas, classroom teachers, principals, and two secondary mathematics teachers developed a set of performance tasks for each grade level K4 (Cowen, Alig, Bannon, Federer, Haas, Nader, Skitzki, Smith, Strachan, Svec & Thornton, 1996, 1997). The performance tasks were called Snapshots because each task provided a glimpse of student growth at certain times throughout the school year. The topics of time, money, area, length, volume and chance were used as a framework to create the performance tasks and guide daily instruction (see Appendix 1). Teachers were expected to select lessons that helped student performance on these tasks. These topics and their tasks ascended sophistication throughout K-4. In order to capture and assess student growth over time, students were given the identical performance tasks two to three times during the school year (September, February, May).
The Snapshot tasks connected instruction and assessment. Prior to each individual performance on the task, teachers and students read and discussed the task. Students determined the important information in the problem and suggested possible strategies for finding solutions. Often, during these mathematical dialogues, students solve the problem. Nevertheless, all students were still expected to solve the problem independently (i.e., defending their thinking and solutions with illustrations, words and calculations). What follow are some of the teacher-designed performance tasks and the students' solutions to these problems. For an entire set of performance tasks and some examples of students' solutions to the tasks, see Appendix 1.
LENGTH: (Performance Task)
Third Grade: Every day Sean took Rex, his grandmother's dog, for a long walk around the nearby high school. The total distance they walked each day was 3-1/2 kilometers. Erica, Sean's cousin, lived next door to their grandmother. Erica took her dog, Spot, to long walks four times a week. They walked 5-1/4 kilometers each time. Who walked the farthest by the end of each week?
Teachers noticed the following about one student's responses:
* October performance: Student constructs two calendars for each dog-walker. On one calendar seven days were marked off. Underneath each day was the number 3-1/2. The second calendar had four days with 5-1/4 listed under Monday, Tuesday, Wednesday and Friday. Student calculated the time on each calendar separately. Column addition was performed on whole numbers and then on fractions. …